The Fan Theorem, its strong negation, and the determinacy of games

Abstract

In the context of a weak formal theory called Basic Intuitionistic Mathematics $\textsf{BIM}$, we study Brouwer’s Fan Theorem and a strong negation of the Fan Theorem, Kleene’s Alternative (to the Fan Theorem). We prove that the Fan Theorem is equivalent to contrapositions of a number of intuitionistically accepted axioms of countable choice and that Kleene’s Alternative is equivalent to strong negations of these statements. We discuss finite and infinite games and introduce a constructively useful notion of determinacy. We prove that the Fan Theorem is equivalent to the Intuitionistic Determinacy Theorem. This theorem says that every subset of Cantor space $2^\omega$ is, in our constructively meaningful sense, determinate. Kleene’s Alternative is equivalent to a strong negation of a special case of this theorem. We also consider a uniform intermediate value theorem and a compactness theorem for classical propositional logic. The Fan Theorem is equivalent to each of these theorems and Kleene’s Alternative is equivalent to strong negations of them. We end with a note on ‘stronger’ Fan Theorems. The paper is a sequel to Veldman (Arch Math Logic 53:621–693, 2014).

1 Introduction

1.1 Intuitionistic reverse mathematics

L.E.J. Brouwer did not present his intuitionistic mathematics as a formal axiomatic theory. He did not like formalism and formalization and anxiously maintained the distinction between a mathematical proof and the linguistic expression that should help us to recover the proof but may fail to do so. The challenge to develop formal theories coming close to Brouwer’s intentions was taken up by A. Heyting, G. Gentzen, S.C. Kleene, G. Kreisel, J. Myhill, A.S. Troelstra, and others.

Given a preferably weak formal basic theory $\Gamma$ and a formal proof in $\Gamma$ of a statement T from some extra assumption A, one may ask: is there also a formal proof in $\Gamma$ of this extra assumption A from the statement T? The study of such questions, as far as they belong to the field of classical analysis or second-order arithmetic, is called reverse mathematics, see [27]. The weak basic theory there is $\textsf{RCA}_0$.

1.2 The basic theory $\textsf{BIM}$

Our subject is intuitionistic reverse mathematics.

The weak basic theory we use is the two-sorted first-order intuitionistic theory $\textsf{BIM}$ (Basic Intuitionistic Mathematics), introduced in [44]. The domain of discourse of $\textsf{BIM}$ consists of two kinds of objects: natural numbers and infinite sequences of natural numbers. The axioms express some basic assumptions like the (full) principle of induction on the set $\omega$ of the natural numbers, and the fact that the set $\omega ^\omega$ of the infinite sequences of natural numbers is closed under the recursion-theoretic operations.

The reason that we use a basic theory different in spirit from the basic theory used in classical reverse mathematics is that, in intuitionistic analysis, one prefers the notion of an infinite sequence of natural numbers as a primitive notion above the notion of a subset of the set of the natural numbers, see [44, Sect. 5].

For the intuitionistic mathematician, the set $\omega ^\omega$ of all infinite sequences of natural numbers is not, as one sometimes says when explaining the notion of ‘set’ that lies at the basis of classical set theory, the result of taking together the earlier constructed and completed items that are to be the ‘elements’ of the set. The set $\omega ^\omega$ is a realm of possibilities: it is a framework for constructing, in the future, in all kinds of possibly as yet unforeseen ways, the objects that will be called the elements of the set. There are several kinds of infinite sequences $\alpha = \bigl (\alpha (0), \alpha (1), \ldots \bigr )$ of natural numbers. Sometimes, $\alpha$ is the result of executing a program, a finitely formulated algorithm. It is also possible that $\alpha$ is the result of a more or less free step-by-step construction that is not governed by a rule formulated at the start.

1.3 Two interpretations

The axioms of $\textsf{BIM}$ hold for their intended interpretation, the interpretation given to them by the intuitionistic mathematician. The axioms of $\textsf{BIM}$ also become true for her if she assumes that the elements of $\omega ^\omega$ are just the Turing-computable functions from $\omega$ to $\omega$. Turing-computable functions may be represented by the natural number coding their program, and $\textsf{BIM}$ may be seen to be a conservative extension of first-order intuitionistic arithmetic $\textsf{HA}$, Heyting arithmetic.

The model given by the computable functions thus is the second interpretation of $\textsf{BIM}$. Our study of this model is a contribution to intuitionistic recursive analysis.

In the weak context given by $\textsf{BIM}$ one may study the further assumptions of the intuitionistic mathematician. They fall into three groups:

• (1) Axioms of Countable Choice,

• (2) Brouwer’s Continuity Principle and the Axioms of Continuous Choice, and

• (3) Brouwer’s Thesis on bars in Baire space $\omega ^\omega$ and the Fan Theorem.

The intuitionistic mathematican is prepared to argue the plausibility of these assumptions for her intended interpretation.

She defends the Axioms of Countable Choice, for instance, by explaining that the functions promised by the axioms may be constructed step by step.

She has no argument for the truth of the further assumptions under the second interpretation, where every function is assumed to be given by an algorithm. It is not clear to her if the Axioms of Countable Choice then are true.

Brouwer’s Continuity Principle and its extensions, the Axioms of Continuous Choice, certainly fail in the second interpretation.

The Thesis on bars in $\omega ^\omega$ was introduced by Brouwer for the sake of the Fan Theorem. The Fan Theorem itself, dating from 1924, see [5], might be called the Thesis on Bars in Cantor space $2^\omega$, see [40, 41], and [44, Sect. 2.3].

In 1950, see [12], Kleene saw that, in our second interpretation, also the Fan Theorem, and, a fortiori, the Thesis on bars in $\omega ^\omega$, do not hold. Actually, a positively formulated strong negation of the Fan Theorem becomes true. In [44], we called this statement Kleene’s Alternative (to the Fan Theorem).

1.4 Strong negations

The (strict) Fan Theorem, ${\textbf {FT}}$, see Sect. 2.2.4, is the statement¹

$$\begin{aligned} \forall \alpha [Bar_{2^\omega }(D_\alpha ) \rightarrow \exists n[ Bar_{2^\omega }(D_{{\overline{\alpha }} n})]], \end{aligned}$$

and Kleene’s Alternative (to the Fan Theorem), ${\textbf {KA}}$, see Sect. 2.2.8, is the statement

$$\begin{aligned} \exists \alpha [Bar_{2^\omega }(D_\alpha ) \;\wedge \; \forall n[ \lnot Bar_{2^\omega }(D_{{\overline{\alpha }} n})]]. \end{aligned}$$

We want to call ${\textbf {KA}}$ the strong negation $\lnot {\textbf {! FT}}$ of ${\textbf {FT}}$. In general, if we decide to call a statement B the strong negation of a statement A, B will be a statement more positive than the negation of A that constructively implies the negation of A. We do not require that the statement B is completely positive in the sense that the corresponding formula does not contain $\lnot$ and $\rightarrow$.² We do not introduce strong negation as a syntactical operation on formulas. It is important to realize that, once we have understood that statement A is equivalent to statement B, it may be the case that statements C, D, which have been chosen to be called the strong negations of A, B, respectively, fail to be equivalent.

If we have decided to call the formula (denoted by) B the strong negation of the formula (denoted by) A, we will write $B=\lnot !A$, but note that this notation belongs to the meta-language of $\textsf{BIM}$. $\lnot !$ is neither a connective belonging to the language of $\textsf{BIM}$ nor a syntactical operation on formulas.

We shall prove a number of results of the form:

In $\textsf{BIM}$, A is equivalent to B and $\lnot ! B$ is equivalent to $\lnot ! A$.

When we do so, we try to explain that the conclusions $A\rightarrow B$ and $\lnot ! B\rightarrow \lnot ! A$ have a common ground and that also the conclusions $B\rightarrow A$ and $\lnot ! A\rightarrow \lnot ! B$ have a common ground.

1.5 Contrapositions or reversals

We may compare Weak König’s Lemma, ${\textbf {WKL}}$, see 2.2.12:

$$\begin{aligned} \forall \alpha [\forall n[\lnot Bar_{2^\omega }(D_{{\overline{\alpha }} n})]\rightarrow \exists \gamma \in 2^\omega \forall n [ \alpha (\overline{\gamma } n) = 0 ]]. \end{aligned}$$

with the (strict) Fan Theorem, ${\textbf {FT}}$:

$$\begin{aligned} \forall \alpha [Bar_{2^\omega }(D_\alpha ) \rightarrow \exists n[ Bar_{2^\omega }(D_{{\overline{\alpha }} n})]]. \end{aligned}$$

We like to say that ${\textbf {WKL}}$ is a reversal or contraposition of ${\textbf {FT}}$ and also that ${\textbf {FT}}$ is a reversal or contraposition of ${\textbf {WKL}}$.

We like to write: ${\textbf {WKL}}=\overleftarrow{{\textbf {FT}}}$ and ${\textbf {FT}}=\overleftarrow{{\textbf {WKL}}}$.

In general, if we call a statement B the contraposition or reversal $\overleftarrow{A}$ of a statement A, both A and B will be largely positively formulated statements and the classical mathematician would think A and B are equivalent, but, constructively, A and B will have quite different meanings.

This is clear from the above example as ${\textbf {FT}}$ is intuitionistically true (under our first interpretation) and ${\textbf {WKL}}$ is false (in both interpretations).

It may happen also that both A and B are intuitionistically true (at least under the first interpretation), although they make different sense. An important example of this phenomenon is given by $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, see Sect. 4.7, and $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}}$, see Sect. 5.4 and Lemma 5.3.

Note that Kleene’s Alternative (to the Fan Theorem), ${\textbf {KA}}$, might be called the strong negation $\lnot !{\textbf {WKL}}$ of ${\textbf {WKL}}$ as well as the strong negation $\lnot !{\textbf {FT}}$ of ${\textbf {FT}}$.

We do not claim that, given a statement A, there always is a unique candidate for being called the contraposition of A. We do not introduce contraposition as a syntactical operation on formulas and use the term only in certain specific cases. It is important to realize that, once we have understood that statement A is equivalent to statement B, it may be the case that statements C, D, which one would like to call contrapositions of A, B, respectively, fail to be equivalent.

1.6 Non-intuitionistic assumptions

The reader may wonder why we pay attention to statements that fail in both our models, like Weak König’s Lemma, ${\textbf {WKL}}$, and Bishop’s Omniscience Principles, ${\textbf {LPO}}$, see 2.2.15, and ${\textbf {LLPO}}$, see 2.2.14. Doing so, however, we come to understand that certain other statements, being equivalent, in $\textsf{BIM}$, to one of them, also do not make sense in either one of our two interpretations.

1.7 Our aim

As in [43] and [44], it is our aim, in this paper, to find statements that are, in $\textsf{BIM}$, equivalent to either ${\textbf {FT}}$ or $\lnot !{\textbf {FT}}={\textbf {KA}}$.

1.8 The contents of the paper

Apart from this introduction, the paper contains Sects. 2–13.

Section 2 is divided into two Subsections. In Sect. 2.1 we introduce the formal system $\textsf{BIM}$. Section 2.2 lists a number of assumptions one might study in the context of $\textsf{BIM}$.

In Sect. 3 we prove that, in $\textsf{BIM}$, the $\varvec{\Sigma }^0_1$-Separation Principle $\varvec{\Sigma }^0_1$-${\textbf {Sep}}$ is equivalent to ${\textbf {WKL}}$.

In Sect. 4 we formulate some special cases of the First Axiom of Choice ${\textbf {AC}}_{\omega ,\omega }$, among them $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, the $\varvec{\Pi }^0_1$-Axiom of Countable Binary Choice.

In Sect. 5 we introduce $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}}$, a contraposition of $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, and we prove that, in $\textsf{BIM}$, $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}}$ is equivalent to ${\textbf {FT}}$, and a strong negation of $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}}$ is equivalent to $\lnot !{\textbf {FT}}={\textbf {KA}}$.

Section 5 thus shows that a contraposition of $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ fails in our second interpretation. This gives us no conclusion about the validity of $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ itself in our second interpretation.

In Sect. 6 we formulate some special cases of the Second Axiom Scheme of Countable Choice ${\textbf {AC}}_{\omega ,\omega ^\omega }$, among them $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2^\omega }$, the $\varvec{\Pi }^0_1$-Axiom of Countable Compact Choice.

In Sect. 7 we introduce $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2^\omega }}$, a contraposition of $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2^\omega }$, and we prove that, in $\textsf{BIM}$, $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2^\omega }}$ is equivalent to ${\textbf {FT}}$, and a strong negation of $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2^\omega }}$ is equivalent to $\lnot !{\textbf {FT}}$.

In Sect. 8 we introduce $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$, a contraposition of a statement provable in $\textsf{BIM}$, to wit, the $\varvec{\Pi }^0_1$-“axiom” of Twofold Compact Choice.

We prove that, in $\textsf{BIM}+\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$ is equivalent to ${\textbf {FT}}$. There is no companion result for $\lnot !{\textbf {FT}}$.

In Sect. 9 we consider finite and infinite games. We explain in what sense we want to call such games I-determinate or II-determinate. We see that $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}}$ can be read as the statement that certain 2-move games are I-determinate. We prove: in $\textsf{BIM}$, ${\textbf {FT}}$ is equivalent to the statement that every subset of Cantor space $2^\omega$ is (weakly) I-determinate, and $\lnot !{\textbf {FT}}$ is equivalent to the statement that there exists an open subset of $2^\omega$ that positively fails to be I-determinate.

In Sect. 10 we consider a Uniform Contrapositive Intermediate Value Theorem $\overleftarrow{{\textbf {UIVT}}}$ and we prove: in $\textsf{BIM}$, ${\textbf {FT}}$ is equivalent to $\overleftarrow{{\textbf {UIVT}}}$ and $\lnot !{\textbf {FT}}$ is equivalent to a strong negation of $\overleftarrow{{\textbf {UIVT}}}$.

In Sect. 11 we see that, if one formulates the compactness theorem for classical propositional logic carefully and contrapositively, one obtains a statement that, in $\textsf{BIM}$, is equivalent to ${\textbf {FT}}$. $\lnot !{\textbf {FT}}$ is equivalent to a strong negation of this statement.

In Sect. 12 we ask the reader’s attention for the Approximate-Fan Theorem ${\textbf {AppFT}}$, a statement stronger than ${\textbf {FT}}$. We did so already in [44, Sect. 10.2], see also [45]. ${\textbf {AppFT}}$ is studied further in [45].

Section 13 contains a list of defined notions. This section may be used as a reference by the reader of the preceding sections.

2 The formal system $\textsf{BIM}$

2.1 The basic axioms

$\textsf{BIM}$, introduced in [44, Section 6], has two kinds of variables, numerical variables $m,n,p,\ldots$, whose intended range is the set $\omega$ of the natural numbers, and function variables $\alpha , \beta , \gamma , \ldots$, whose intended range is the set $\omega ^\omega$ of all infinite sequences of natural numbers. There is a numerical constant 0. There are five unary function constants: Id, a name for the identity function, $\underline{0}$, a name for the zero function, S, a name for the successor function, and K, L, names for the projection functions. There is one binary function symbol J, a name for the pairing function on $\omega$. From these symbols numerical terms are formed in the usual way. The basic terms are the numerical variables and the numerical constant and other terms are obtained from earlier constructed terms by the use of a function symbol with parentheses indicating function application. The function constants $\textit{Id}$, $\underline{0}$, S, K and L and the function variables are the only function terms.

$\textsf{BIM}$ has two equality symbols, $=_0$ and $=_1$. The first symbol may be placed between numerical terms and the second one between function terms. When confusion seems improbable we simply write $=$ and not $=_0$ or $=_1$. The usual axioms for equality are part of $\textsf{BIM}$. A basic formula is an equality between numerical terms or an equality between function terms. A basic formula in the strict sense is an equality between numerical terms. We obtain the formulas of the theory from the basic formulas by using the connectives, the numerical quantifiers and the function quantifiers.

The logic of the theory is intuitionistic logic.

Our first axiom is

Axiom 1

(Axiom of Extensionality)

$$\begin{aligned}\forall \alpha \forall \beta [ \alpha =_1 \beta \leftrightarrow \forall n [ \alpha (n) =_0 \beta (n) ] ] \end{aligned}$$

The Axiom of Extensionality guarantees that every formula will be provably equivalent to a formula built up by means of connectives and quantifiers from basic formulas in the strict sense.

The second axiom is the axiom on the unary function constants Id, $\underline{0}$, S, K, L, and the binary function constant J.

Axiom 2

$$\begin{aligned}&\displaystyle \forall n[Id(n) = n] \wedge \\ {}&\displaystyle \forall n [ \lnot (S(n) = 0) ] \;\wedge \; \forall m \forall n [ S(m) = S(n) \rightarrow m = n ]\;\wedge \\&\displaystyle \forall n [ \underline{0}(n) = 0]\;\wedge \\&\displaystyle \forall m \forall n [ K\bigl (J(m,n)\bigr ) = m \;\wedge \; L\bigl (J(m,n)\bigr ) = n \;\wedge \; J\bigl (K(n),L(n)\bigr )=n]\end{aligned}$$

Thanks to the presence of the pairing function we may treat binary, ternary and other non-unary operations on $\omega$ as unary functions. “$\alpha (m,n,p)$”, for instance, will be an abbreviation of “$\alpha \bigl (J(J(m,n),p)\bigr )$”.

We also introduce the following notation: for each n, $n':= K(n)$ and $n'':= L(n)$, and, for all m, n,  $(m,n):= J(m,n)$. The last part of Axiom 2 now reads as follows: $\forall m\forall n[(m,n)'=m \;\wedge \; (m,n)''=n \;\wedge \; (n',n'')=n]$.

The next axiom³ asks for the closure of the set $\omega ^\omega$ under composition, pairing, primitive recursion and unbounded search.

Axiom 3

$$\begin{aligned}{} & {} \forall \alpha \forall \beta \exists \gamma \forall n [ \gamma (n) = \alpha \bigl (\beta (n)\bigr ) ]\;\wedge \\ {}{} & {} \forall \alpha \forall \beta \exists \gamma \forall n[\gamma (n) = \bigl (\alpha (n), \beta (n)\bigr )]\;\wedge \\{} & {} \forall \alpha \forall \beta \exists \gamma \forall m \forall n [ \gamma (m,0)= \alpha (m) \wedge \gamma \bigl (m,S(n)\bigr ) = \beta \bigl (m,n,\gamma (m,n)\bigr ) ]\;\wedge \\{} & {} \forall \alpha [ \forall n \exists m [ \alpha (n,m) = 0 ] \rightarrow \exists \gamma \forall n [ \alpha \bigl (n, \gamma (n)\bigr ) = 0 ] ] \end{aligned}$$

We also need the Axiom Scheme of Induction:

Axiom 4

$$\begin{aligned}(P(0) \wedge \forall n [ P(n) \rightarrow P\bigl (S(n)\bigr )]) \rightarrow \forall n [P(n)]\end{aligned}$$

Instances of this axiom scheme are obtained by substituting a formula $\phi =\phi (m_0, m_1, \ldots , m_{k-1}, \alpha _0,\alpha _1, \ldots , \alpha _{l-1}, n)$ for P and taking the universal closure of the resulting formula. The reader should understand the further axiom schemes we are to mention in this paper in the same way.

The axioms 1-4, together with the usual axioms for equality, define the system $\textsf{BIM}$.

Note $\textsf{BIM}$ has the full induction scheme whereas $\textsf{RCA}_0$ has only $\Sigma ^0_1$-induction, see [27, Definition II.1.5], a fact that is indicated by the suffix $_0$. We did not study the possibility of restricting induction likewise and we do not answer the question if our results might have been obtained in a system that probably should be called $\textsf{BIM}_0$.

We form a conservative extension of $\textsf{BIM}$ by adding constants for all primitive recursive functions and relations and making their defining equations into axioms. Primitive recursive relations are present via their characteristic functions. ‘$x<y$’, for instance, will be short for: ‘$\chi _<(x,y)\ne 0$’. Somewhat loosely, we also denote this conservative extension of $\textsf{BIM}$ by the acronym $\textsf{BIM}$ although one might decide to use the acronym $\textsf{BIM}^+$, see [31].

$\textsf{BIM}$ may be compared to the system H introduced in[9] and to the system ${\textbf {EL}}$ occurring in[30] and to the system ${\textbf {IRA}}$, proposed by J.R. Moschovakis and G. Vafeiadou, see [23]. A precise proof of the fact that $\textsf{BIM}$ and these systems are essentially equivalent may be found in [31].

2.2 Possible further assumptions

2.2.1 First axiom scheme of countable choice, ${\textbf {AC}}_{\omega ,\omega } (={\textbf {AC}}_{0,0})$:

$$\begin{aligned} \forall n\exists m[R(n,m)] \rightarrow \exists \gamma \forall n[R\bigl (n,\gamma (n)\bigr )]. \end{aligned}$$

The intuitionistic mathematician accepts ${\textbf {AC}}_{\omega ,\omega }$. If $\forall n\exists m[R(n,m)]$, she builds the promised $\gamma$ step by step, first choosing $\gamma (0)$ such that $R\bigl (0,\gamma (0)\bigr )$, then choosing $\gamma (1)$ such that $R\bigl (1,\gamma (1)\bigr )$, and so on. In her view, there is no need to give a finite description or algorithm that determines the infinitely many values of $\gamma$ at one stroke.⁴

2.2.2 Second axiom scheme of countable choice, ${\textbf {AC}}_{\omega ,\omega ^\omega }={\textbf {AC}}_{0,1}$:

$$\begin{aligned} \forall n \exists \gamma [R(n,\gamma )]\rightarrow \exists \gamma \forall n[R(n, \gamma ^{\upharpoonright n} )]. \end{aligned}$$

The intuitionistic mathematician accepts ${\textbf {AC}}_{\omega , \omega ^\omega }$. If $\forall n \exists \gamma [R(n, \gamma )]$, she first starts a project for building $\gamma ^{\upharpoonright 0}$ with the property $R(0,\gamma ^{\upharpoonright 0})$ and determines $\gamma ^{\upharpoonright 0}(0)$, she then starts a project for building $\gamma ^{\upharpoonright 1}$ with the property $R(1,\gamma ^{\upharpoonright 1})$ and determines $\gamma ^{\upharpoonright 1}(0)$, and also, continuing the project started earlier, $\gamma ^{\upharpoonright 0}(1)$, she then starts a project for building $\gamma ^{\upharpoonright 2}$ with the property $R(2,\gamma ^{\upharpoonright 2})$ and determines $\gamma ^{\upharpoonright 2}(0)$ and also, continuing the projects started earlier, $\gamma ^{\upharpoonright 1}(1)$ and $\gamma ^{\upharpoonright 0}(2)$, $\ldots$.

2.2.3 The Fan Theorem as an axiom scheme, ${\textbf {FAN}}$:

$$\begin{aligned} \forall \beta [\bigl (Fan(\beta )\;\wedge \;Bar_{\mathcal {F}_\beta }(B)\bigr )\rightarrow \exists a[D_a\subseteq B\;\wedge \;Bar_{\mathcal {F}_\beta }(D_a)]]. \end{aligned}$$

$\beta$ is an explicit fan-law if and only if $Fan(\beta )$ and, in addition, $\exists \gamma \forall s\forall m[\beta (s*\langle m \rangle )=0\rightarrow m\le \gamma (s)]$. We then write $Fan^+(\beta )$. If $Fan^+(\beta )$, then $\mathcal {F}_\beta$ is called an explicit fan.

Lemma 2.1

$\textsf{BIM}\vdash \forall \beta [Fan^+(\beta )\leftrightarrow \bigl (Spr(\beta )\;\wedge \;\exists \gamma \forall n\forall s\in \omega ^n[\beta (s)=0\rightarrow s\le \gamma (n)]\bigr )]$.

Proof

Let $\beta$ be given such that $Fan^+(\beta )$. Find $\gamma$ such that $\forall s \forall m[\beta (s*\langle m \rangle )=0\rightarrow m\le \gamma (s)]$. Define $\delta$ such that $\delta (0)=0=\langle \;\rangle$, and, for each n, $\delta (n+1) = \max (\{s*\langle m\rangle \mid \beta \bigl (s*\langle m \rangle )=0 \;\wedge \; s\le \delta (n)\;\wedge \;m\le \gamma (s)\}\bigr )$. One proves by induction that $\forall n\forall s\in \omega ^n[\beta (s)=0\rightarrow s\le \delta (n)]$.

Conversely, let $\beta , \gamma$ be given such that $Spr(\beta )\;\wedge \;\forall n\forall s\in \omega ^n[\beta (s)=0\rightarrow s\le \gamma (n)]$. Define $\delta$ such that, for each n, for each s in $\omega ^n$ such that $\beta (s)=0$, $\delta (s)=\max \bigl (\{m\mid \beta (s*\langle m\rangle )=0\;\wedge \; s*\langle m \rangle \le \gamma (n+1) \}\bigr )$. Note that $\forall s \forall m[\beta (s*\langle m\rangle )=0\rightarrow m\le \delta (s)]$ and conclude that $Fan^+(\beta )$. $\square$

2.2.4 The (strict) Fan Theorem, ${\textbf {FT}}$:

$$\begin{aligned}{} & {} \forall \alpha [\textit{Bar}_{2^\omega }(D_\alpha ) \rightarrow \exists m[\textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} m})]], \;or, \;equivalently,\\{} & {} \forall \beta [Fan^+(\beta )\rightarrow \forall \alpha [\textit{Bar}_{\mathcal {F}_\beta }(D_\alpha ) \rightarrow \exists m[\textit{Bar}_{\mathcal {F}_\beta }(D_{{\overline{\alpha }} m})]]]\;\textit{or,}\;\textit{equivalently},\\{} & {} \forall \beta [Fan^+(\beta )\rightarrow \forall \alpha [\textit{Bar}_{\mathcal {F}_\beta }(D_\alpha ) \rightarrow \exists m\forall \gamma \in \mathcal {F}_\beta \exists n\le m[{\overline{\gamma }} n \in D_\alpha ]]]. \end{aligned}$$

Theorem 2.2

$\textsf{BIM}\vdash {\textbf{FT}}\leftrightarrow$

$\forall \alpha [\bigl (Thinbar_{2^\omega }(D_\alpha )\;\wedge \; D_\alpha \subseteq 2^{<\omega }\bigr )\rightarrow \exists n\forall m>n[m\notin D_\alpha ]]$.

Proof

The proof is left to the reader. $\square$

2.2.5 The strengthened (strict) Fan Theorem, ${\textbf {FT}}^+$:

Theorem 2.3

$\textsf{BIM}\vdash \textbf{FT}^+\leftrightarrow \forall \beta [Fan(\beta )\rightarrow \forall \alpha [\bigl (Thinbar_{\mathcal {F}_\beta }(D_\alpha )\;\wedge \; \forall s \in D_\alpha [\beta (s)=0]\bigr )\rightarrow \exists n\forall m>n[m\notin D_\alpha ]]]$.

Proof

The proof is left to the reader. $\square$

Note that $\textsf{BIM}+{\textbf {FAN}}\vdash {\textbf {FT}}^+$.

2.2.6 Brouwer’s thesis: bar induction as an axiom scheme, ${\textbf {BARIND}}$:

$$\begin{aligned}\bigl (Bar_{\omega ^\omega }(B) \;\wedge \; \forall s[s\in B\rightarrow s\in E]\;\wedge \;\forall s[\forall n[s*\langle n \rangle \in E]\leftrightarrow s\in E]\bigr )\rightarrow \langle \;\rangle \in E.\end{aligned}$$

$E\subseteq \omega$ is called inductive if and only if $\forall s[\forall n[s*\langle n \rangle \in E]\rightarrow s\in E]$ and monotone if and only if $\forall s\forall n[s\in E\rightarrow s*\langle n \rangle \in E]$.

Brouwer derived ${\textbf {FAN}}$ from ${{\textbf {BARIND}}}$, see [44, Sects. 2.2 and 2.3]. We repeat the proof, in order to prepare the reader for Theorem 12.4.

Theorem 2.4

$\textsf{BIM}+\textbf{BARIND}\vdash \textbf{FAN}$.

Proof

Let $\beta$ be given such that $Fan(\beta )$ and $\beta (\langle \;\rangle )=0$.⁵ Assume $Bar_{\mathcal {F}_\beta }(B)$. Define $B':=B\cup \{s\mid \beta (s)\ne 0\}$. We will prove that $Bar_{\omega ^\omega }(B')$. Let $\gamma$ be given. Define $\gamma ^*$ such that, for each n, if $\beta \bigl ({\overline{\gamma }}(n+1)\bigr )=0$, then $\gamma ^*(n)=\gamma (n)$, and, if not, then $\gamma ^*(n)=\mu p[\beta (\overline{\gamma ^*} n*\langle p\rangle )=0]$. Note that $\gamma ^*\in \mathcal {F}_\beta$ and find n such that $\overline{\gamma ^*}n\in B$. Either ${\overline{\gamma }} n=\overline{\gamma ^*}n$ and ${\overline{\gamma }} n \in B$ or ${\overline{\gamma }} n\ne \overline{\gamma ^*}n$ and $\beta ({\overline{\gamma }} n)\ne 0$. In both cases, ${\overline{\gamma }} n \in B'$. We thus see that $\forall \gamma \exists n[{\overline{\gamma }} n \in B']$, i.e. $Bar_{\omega ^\omega }(B')$.

Let E be the set of all s such that either $\beta (s)\ne 0$ or $\beta (s) = 0$ and $\exists a[ D_a \subseteq B \;\wedge \; Bar_{\mathcal {F}_\beta \cap s}(D_a)]$.

For every s, if $\beta (s) =0$ and $s\in B$, define $a:=\overline{\underline{0}}s*\langle 1 \rangle$ and note that $\{s\}=D_a\subseteq B$ and $Bar_{\mathcal {F}_\beta \cap s}(D_a)$. Conclude that $B\subseteq E$.

Now let s be given such that $\forall m[s*\langle m \rangle \in E]$. Find n such that $\forall m\ge n[\beta (s*\langle m\rangle ) \ne 0]$. Find b in $\omega ^n$ such that $\forall m<n[\beta (s*\langle m \rangle )=0 \rightarrow \bigl (D_{b(m)}\subseteq B\;\wedge \;Bar_{\mathcal {F}_\beta \cap s*\langle m \rangle }(D_{b(m)})\bigr )]$ and $\forall m<n[\beta (s*\langle m \rangle )\ne 0\rightarrow b(m)=\langle \;\rangle ]$. Find a such that $p:=length(a)=\max _{m<n}length\bigl (b(m)\bigr )$ and $\forall t<p[a(t)\ne 0\leftrightarrow \exists m<n[\bigl (b(m)\bigr )(t)\ne 0]]$. Note that $D_a\subseteq B$ and $Bar_{\mathcal {F}_\beta \cap s}(D_a)$, and conclude that $s\in E$. We thus see that $\forall s[\forall m[s*\langle m \rangle \in E]\rightarrow s \in E]$, i.e. E is inductive.

Note that $\forall s\forall m[s\in E\rightarrow s*\langle m \rangle \in E]$, i.e. E is monotone.

Using ${\textbf {BARIND}}$, conclude that $\langle \;\rangle \in E$, i.e. $\exists a[ D_a \subseteq B \;\wedge \; Bar_{\mathcal {F}_\beta }(D_a)]$. $\square$

2.2.7 Church’s thesis, ${\textbf {CT}}$:

$$\begin{aligned}\exists \tau \exists \psi \forall \alpha \exists e\forall n \exists z[z=\mu i[\tau (e, n, i )\ne 0] \;\wedge \; \psi (z) = \alpha (n)].\end{aligned}$$

Kleene has shown that ${\textbf {CT}}$ is true in the model of $\textsf{BIM}$ given by the computable functions. He provided Kálmar-elementary functions $\tau ,\psi$ satisfying the above conditions. Note that our formulation of ${\textbf {CT}}$ is cautious and somewhat weaker than the usual one. We do not require that the set $\{(e,n,z)\mid \tau (e,n,z) \ne 0\}$ coincides with Kleene’s set T, but only ask that the set behaves appropriately. A similar ‘abstract’ approach to Church’s Thesis has been advocated by F. Richman, see [25] and [3, Chapter 3, Sect. 1].

2.2.8 Kleene’s alternative (to the Fan Theorem), $\lnot !{\textbf {FT}}$:

$$\begin{aligned} \exists \alpha [ \textit{Bar}_{2^\omega }(D_\alpha )\;\wedge \; \forall m [ \lnot \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} m})]].\end{aligned}$$

Theorem 2.5

$\textsf{BIM} + \textbf{CT} \vdash \lnot !\textbf{FT}$.

Proof

Let $\tau , \psi$ be as in ${\textbf {CT}}$. Define $\alpha$ such that, for all m, for all c in $2^{<\omega }$ such that $length(c)=m$,

$$\begin{aligned} \alpha (c) = 0\leftrightarrow \forall e<m\forall z < m[ z=\mu i[ \tau ( e, e, i)\ne 0]\rightarrow c(e) = 1-\psi (z)]. \end{aligned}$$

Let $\gamma$ in $2^\omega$ be given. Find e such that, for all n, $\gamma (n) = \psi (\mu i[\tau ( e,n,i)\ne 0])$. Define $z=\mu i[\tau ( e, e, i )\ne 0]$. Note that $\gamma (e)=\psi (z)$. Define $m:= \max (e, z) +1$ and note that $\alpha ({\overline{\gamma }} m) \ne 0$. Conclude that $\forall \gamma \in 2^\omega \exists m[\alpha ({\overline{\gamma }} m)\ne 0]$, i.e. $\textit{Bar}_{2^\omega }(D_\alpha )$.

Let m be given. Find c in $2^{<\omega }$ such that $length(c)=m$ and $\forall e< m\forall z <m[z=\mu i[\tau ( e, e, i)\ne 0]\rightarrow c(e) = 1-\psi (z)]$. Note that $\forall n \le m[\alpha ({\overline{c}} n) = 0]$. Define $\gamma :=c*\underline{0}$ and note that $\forall n [c*\underline{{\overline{0}}} n > m]$ and conclude that $\lnot \exists k[{\overline{\gamma }} k<m \;\wedge \;\alpha ({\overline{\gamma }} k)\ne 0]$ and $\lnot \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} m})$. Conclude that $\forall m[\lnot Bar_{2^\omega }(D_{{\overline{\alpha }} m})]$. $\square$

Theorem 2.5 is due to Kleene, see [12, §3] and [13, Lemma 9.8]. There is a proof in [30, vol. I, Chapter 4, Sect. 7.6]. In [44, Sect. 3] one finds two more proofs.

We do not know the answer to the question if $\textsf{BIM}+\lnot !{\textbf {FT}}\vdash {\textbf {CT}}$.

2.2.9 Brouwer’s (unrestricted) continuity principle as an axiom scheme, ${\textbf {BCP}}$:

$$\begin{aligned} \forall \alpha \exists n[\alpha R n]\rightarrow \forall \alpha \exists m \exists n \forall \beta [ {\overline{\alpha }} m \sqsubset \beta \rightarrow \beta R n]. \end{aligned}$$

Brouwer used this principle for the first time in 1918, see [4, Sect. 1, page 13]. The intuitionistic mathematician believes the axiom to be plausible. She argues as follows. If $\forall \alpha \exists n[\alpha Rn]$, I must be able, given any $\alpha$, to find effectively an n as promised, also if the values of $\alpha$ are disclosed one by one and nothing is told about a law governing the development of $\alpha$ as a whole. I will decide upon n at some point of time and, at that point of time, only finitely many values of $\alpha$, say $\alpha (0), \alpha (1), \ldots ,\alpha (m-1)$, will be known to me. The number n will satisfy any infinite sequence that is a continuation of $\alpha (0), \alpha (1), \ldots ,\alpha (m-1)$.

The Continuity Principle is revolutionary and changes one’s mathematical perspective, see [36].

The classical mathematician may ask for a consistency proof for ${\textbf {BCP}}$. Kleene proved, using realizability methods, that his formal system ${\textbf {FIM}}$ for intuitionistic analysis, actually an extension of $\textsf{BIM}+{\textbf {FT}}+{\textbf {BCP}}$, is (simply) consistent, see [13, Chapter II, Sect. 9.2]. Kleene’s consistency proof should convince both the classical mathematician and the constructive mathematician, should the latter accept ${\textbf {BARIND}}$ but be plagued by doubts concerning ${\textbf {BCP}}$. It is not difficult to see that $\textsf{BIM}+ {\textbf {CT}} + {\textbf {BCP}}$ is inconsistent, as it implies $\forall \alpha \exists m\forall \beta [{\overline{\beta }} m = {\overline{\alpha }} m \rightarrow \beta = \alpha ]$, see [30, vol. I, Chapter 4, Theorem 6.7].

The next two axioms strengthen ${\textbf {BCP}}$. Kleene’s consistency proof extends to these stronger forms of the Continuity Principle.

2.2.10 The first axiom scheme of continuous choice, ${\textbf {AC}}_{\omega ^\omega ,\omega } (={\textbf {AC}}_{1,0}$):

$$\begin{aligned} \forall \alpha \exists n[\alpha R n]\rightarrow \exists \varphi :\omega ^\omega \rightarrow \omega \forall \alpha [\alpha R \varphi (\alpha )]. \end{aligned}$$

2.2.11 The second axiom scheme of continuous choice, ${\textbf {AC}}_{\omega ^\omega ,\omega ^\omega } (={\textbf {AC}}_{1,1}$):

$$\begin{aligned} \forall \alpha \exists \beta [\alpha R \beta ]\rightarrow \exists \varphi :\omega ^\omega \rightarrow \omega ^\omega \forall \alpha [\alpha R \varphi |\alpha ]. \end{aligned}$$

2.2.12 Weak König’s Lemma, ${\textbf {WKL}}$:

$$\begin{aligned}{} & {} \forall \alpha [\forall n\exists a \in 2^{<\omega }[length(a)= n\;\wedge \;\forall m\le n[\alpha ({\overline{a}} m)=0]\rightarrow \exists \gamma \in 2^\omega \forall n [ \alpha (\overline{\gamma } n) = 0 ]],\\{} & {} or,\; equivalently, \;\forall \alpha [\forall n[\lnot Bar_{2^\omega }(D_{{\overline{\alpha }} n})]\rightarrow \exists \gamma \in 2^\omega \forall n [ \overline{\gamma } n\notin D_\alpha ]]. \end{aligned}$$

${\textbf {WKL}}$, as a classical theorem, dates from 1927, see [14]. It is a contraposition of ${\textbf {FT}}$ and, from a classical point of view, the two are equivalent. The following result may be found in [10], and also in [15] and [11].

Theorem 2.6

$\textsf{BIM}\vdash$ WKL $\rightarrow$ FT.

Proof

Assume ${\textbf {WKL}}$.

Let $\alpha$ be given such that $\textit{Bar}_{2^\omega }(D_\alpha )$. We intend to prove that $\exists n[Bar_{2^\omega }(D_{{\overline{\alpha }} n})]$. Define $\alpha ^*$ such that $\forall s \in 2^{<\omega }[\alpha ^*(s)= 0 \leftrightarrow \forall t\sqsubseteq s[\alpha (t)= 0]]$. Define $\alpha ^{**}$ such that, for all s in $2^{<\omega }$, $\alpha ^{**}(s)= 0$ if and only if either $\alpha ^*(s)= 0$ or $\lnot \exists t \in 2^{<\omega }[length(t)=length(s)\;\wedge \;\alpha ^*(t)=0]$. Note that, for each n, there exists s in $2^{<\omega }$ such that $length(s)=n$ and $\alpha ^{**}(s)=0$. Applying ${\textbf {WKL}}$, find $\gamma$ in $2^\omega$ such that $\forall n[\alpha ^{**}({\overline{\gamma }} n)=0]$. Find n such that $\alpha ({\overline{\gamma }} n)\ne 0$. Conclude that $\lnot \exists t\in 2^{<\omega }[length(t)=n\;\wedge \;\alpha ^*(t)=0]$ and $\forall \delta \in 2^\omega \exists j\le n[\alpha ({\overline{\delta }} j)\ne 0]$ and $\exists m[Bar_{2^\omega }(D_{{\overline{\alpha }} m})]$.

We thus see that $\forall \alpha [Bar_{2^\omega }(D_\alpha )\rightarrow \exists m[Bar_{2^\omega }(D_{{\overline{\alpha }} m})]]$, i.e. ${\textbf {FT}}$. $\square$

2.2.13 Weak König’s Lemma with a uniqueness condition, ${\textbf {WKL!}}$:

$$\begin{aligned}{} & {} \forall \alpha [\bigl (\forall m[\lnot Bar_2^\omega (D_{{\overline{\alpha }} m})]\;\wedge \\{} & {} \forall \gamma \in 2^\omega \forall \delta \in 2^\omega [\gamma \perp \delta \rightarrow \exists m[\alpha ({\overline{\gamma }} m)\ne 0\;\vee \;\alpha ({\overline{\delta }} m)\ne 0]]\bigr )\rightarrow \exists \gamma \forall n[\alpha ({\overline{\gamma }} n)=0]].\end{aligned}$$

The next two theorems, apart from being of interest in themselves, are useful for the discussion in Sect. 8.1.1.

Theorem 2.7

$\textsf{BIM}\vdash \mathbf{WKL!}\rightarrow \textbf{FT}$.

Proof

⁶ Assume ${\textbf {WKL!}}$.

Let $\alpha$ be given such that $Bar_{2^\omega }(D_\alpha )$. We will prove that $\exists m[Bar_{2^\omega }(D_{{\overline{\alpha }} m})]$. If $\alpha (0)=\alpha (\langle \;\rangle )\ne 0$, then $D_{{\overline{\alpha }} 1}=\{\langle \;\rangle \}$ and $Bar_2^\omega (D_{{\overline{\alpha }} 1})$, and we are done. Now assume that $\alpha (\langle \;\rangle )=0$, i.e. $\langle \;\rangle \notin D_\alpha$. Define $\alpha ^*$ such that $\forall s \in 2^{<\omega }[\alpha ^*(s)= 0 \leftrightarrow \forall t\sqsubseteq s[\alpha (t)= 0]]$. Define $\alpha ^{**}$ such that $\alpha ^{**}(\langle \;\rangle )=0$ and,, for all s in $2^{<\omega }\setminus \{\langle \;\rangle \}$, $\alpha ^{**}(s)= 0$ if and only if either $\alpha ^*(s)= 0$ and $\forall t\in 2^{<\omega }[\bigl (length(t)=length(s)\;\wedge \; \alpha ^*(t)= 0\bigr )\rightarrow s\le _{lex}t]$ or $\lnot \exists t \in 2^{<\omega }[length(t)=length(s) \;\wedge \;\alpha ^*(t)=0]$ and $\exists t [\alpha ^{**}(t)=0 \;\wedge \; s = t*\langle 0\rangle ]$. Using induction, one proves that, for each n, there is exactly one s in $2^{<\omega }$ such that $length(s)=n$ and $\alpha ^{**}(s)=0$. Let $\gamma , \delta$ in $2^\omega$ be given such that $\gamma \perp \delta$. Find n such that ${\overline{\gamma }} n\perp {\overline{\delta }} n$ and note that $\alpha ^{**}({\overline{\gamma }} n)\ne 0\;\vee \; \alpha ^{**}({\overline{\delta }} n)\ne 0$. We thus see that $\forall \gamma \in 2^\omega \forall \delta \in 2^\omega [\gamma \perp \delta \rightarrow \exists n[\alpha ^{**}({\overline{\gamma }} n)\ne 0\;\vee \; \alpha ^{**}({\overline{\delta }} n)\ne 0]]$. Applying ${\textbf {WKL!}}$, find $\gamma$ in $2^\omega$ such that $\forall n[\alpha ^{**}({\overline{\gamma }} n)=0]$. Find n such that $\alpha ({\overline{\gamma }} n)\ne 0$. Conclude that $\lnot \exists t\in 2^{<\omega }[length(t)=n \;\wedge \;\alpha ^*(t)=0]$ and $\forall \delta \in 2^\omega \exists j\le n[\alpha ({\overline{\delta }} j)\ne 0]$ and $\exists m[Bar_{2^\omega }(D_{{\overline{\alpha }} m})]$.

We thus see that $\forall \alpha [Bar_{2^\omega }(D_\alpha )\rightarrow \exists m[Bar_{2^\omega }(D_{{\overline{\alpha }} m})]]$, i.e. ${\textbf {FT}}$. $\square$

Theorem 2.8

$\textsf{BIM}\vdash \textbf{FT}\rightarrow \mathbf{WKL!}$.

Proof

Assume ${\textbf {FT}}$.

Let $\alpha$ be given such that $\forall n [\lnot Bar_{2^\omega }(D_{{\overline{\alpha }} n})]]$ and $\forall \gamma \in 2^\omega \forall \delta \in 2^\omega [\gamma \perp \delta \rightarrow \exists n[\alpha ({\overline{\gamma }} n) \ne 0\;\vee \;\alpha ({\overline{\delta }} n)\ne 0]]$. We will prove that $\exists \gamma \in 2^\omega \forall n[\alpha ({\overline{\gamma }} n)=0]$. Define $\alpha ^*$ such that $\forall s \in 2^{<\omega }[\alpha ^*(s)=0\leftrightarrow \forall t\sqsubseteq s[\alpha (t)=0]]$. Let s in $2^{< \omega }$ be given. Note that $\forall \gamma \in 2^\omega \forall \delta \in 2^\omega \exists k[\alpha ^*(s*\langle 0\rangle *{\overline{\gamma }} k)\ne 0\;\vee \;\alpha ^*(s*\langle 1\rangle *{\overline{\delta }} k)\ne 0]$. Conclude that $\forall \gamma \in 2^\omega \exists k[\alpha ^*(s*\langle 0\rangle *\overline{ \gamma ^{\upharpoonright 0}} k)\ne 0\;\vee \;\alpha ^*(s*\langle 1\rangle *\overline{ \gamma ^{\upharpoonright 1}} k)\ne 0]$. Applying ${\textbf {FT}}$, find m such that $\forall \gamma \in 2^\omega \exists k\le m[\alpha ^*(s*\langle 0\rangle *\overline{ \gamma ^{\upharpoonright 0}} k)\ne 0\;\vee \;\alpha ^*(s*\langle 1\rangle *\overline{ \gamma ^{\upharpoonright 1}} k)\ne 0]$ and $\forall \gamma \in 2^\omega \forall \delta \in 2^\omega \exists k\le m[\alpha ^*(s*\langle 0\rangle *{\overline{\gamma }} k)\ne 0\;\vee \;\alpha ^*(s*\langle 1\rangle *{\overline{\delta }} k)\ne 0]$. Conclude that $\forall c\in 2^{<\omega }\forall d\in 2^{<\omega }[length(c)=length(d)=m \rightarrow \alpha ^*(s*\langle 0\rangle *c)\ne 0\;\vee \;\alpha ^*(s*\langle 1\rangle *d)\ne 0]$. Conclude that $\forall c\in 2^{<\omega }[length(c)=m\rightarrow \alpha ^*(s*\langle 0\rangle *c)\ne 0]\;\vee \;\forall d\in 2^{<\omega }[length(d)=m\rightarrow \alpha ^*(s*\langle 1\rangle *d)\ne 0]$.

This last step is justified by the fact that $\textsf{BIM}$ proves the following scheme⁷:

$\forall n[\forall k<n\forall l<n[A(k)\;\vee \;B(l)]\rightarrow (\forall k<n[A(k)]\vee \forall l<n[B(l)])]$.

Now define $\zeta$ such that, for every s in $2^{<\omega }$, $\zeta (s)=\mu k[\forall c\in 2^{<\omega }[length(c)=k\rightarrow \alpha ^*(s*\langle 0\rangle *c)\ne 0]\;\vee \;\forall d\in 2^{<\omega }[length(d)=k\rightarrow \alpha ^*(s*\langle 1\rangle *d)\ne 0]$. Then define $\delta$ in $2^\omega$ such that, for every s in $2^{<\omega }$, $\delta (s)=1$ if $\forall c\in 2^{<\omega }[length(c)=\zeta (s)\rightarrow \alpha ^*(s*\langle 0\rangle *c)\ne 0]$. Finally, define $\gamma$ such that $\forall n[\gamma (n)=1-\delta ({\overline{\gamma }} n)]$. Using the fact that $\forall n[ \lnot Bar_{2^\omega }(D_{{\overline{\alpha }} n})]$, one may prove, by induction, that, for each n, $\forall \varepsilon \in 2^\omega [\varepsilon \perp {\overline{\gamma }} n \rightarrow \exists p[\alpha ^*({\overline{\varepsilon }} p)\ne 0]]$ and $\forall m\exists p>m[p\in 2^{<\omega }\;\wedge \;{\overline{\gamma }} n\sqsubseteq p \;\wedge \;\alpha ^*(p) =0]$. Conclude that $\forall n[\alpha ^*({\overline{\gamma }} n)=0]$ and $\forall n[\alpha ({\overline{\gamma }} n)=0]$.

We thus see that, for each $\alpha$, if $\forall n[\lnot Bar_{2^\omega }(D_{{\overline{\alpha }} n})]$ and $\forall \gamma \in 2^\omega \forall \delta \in 2^\omega [\gamma \perp \delta \rightarrow \exists n[\alpha ({\overline{\gamma }} n)\ne 0\;\vee \;\alpha ({\overline{\delta }} n)\ne 0]]$, then $\exists \gamma \in 2^\omega \forall n[\alpha ({\overline{\gamma }} n)=0]$, i.e. ${\textbf {WKL!}}$. $\square$

2.2.14 The lesser limited principle of omniscience, ${\textbf {LLPO}}$ ⁸:

$$\begin{aligned}\forall \alpha \exists i<2\forall p[2p+i\ne \mu n[\alpha (n)\ne 0]].\end{aligned}$$

Theorem 2.9

$\textsf{BIM}\vdash \textbf{BCP} \rightarrow \lnot \textbf{LLPO}$.

Proof

Assume ${\textbf {LLPO}}$. Using ${\textbf {BCP}}$, find m,i such that $i<2$ and $\forall \alpha [\overline{\underline{0}}m\sqsubset \alpha \rightarrow \forall p[2p+i\ne \mu n[\alpha (n)\ne 0]]]$. Define $\alpha := \overline{\underline{0}}(2\,m+i)*\langle 1 \rangle *\underline{0}$ and note that $\overline{\underline{0}} m\sqsubset \alpha$ and $2m+i=\mu n[\alpha (n)\ne 0]$. Contradiction. $\square$

Theorem 2.10

⁹$\textsf{BIM} \vdash \textbf{WKL} \rightarrow \textbf{LLPO}$.

Proof

Assume ${\textbf {WKL}}$. Let $\alpha$ be given. Define $\beta$ such that, for every s, $\beta (s) = 0$ if and only if $\exists q\exists i<2[s=\overline{\underline{i}}q \;\wedge \;\forall p[2p+i< q \rightarrow 2p+i\ne \mu n[\alpha (n)\ne 0]]]$. Note that $\forall m\exists i<2\forall q\le m[\beta (\overline{\underline{i}} q)=0]$. Using ${\textbf {WKL}}$, find $\gamma$ such that $\forall n[\beta ({\overline{\gamma }} n)=0]$. Define $i:=\gamma (0)$ and conclude: $\gamma = \underline{i}$ and $\forall p[ 2p+i\ne \mu n[\alpha (n)\ne 0]]$. We thus see that $\forall \alpha \exists i<2\forall p[ 2p+i\ne \mu n[\alpha (n)\ne 0]]$, i.e. ${\textbf {LLPO}}$. $\square$

Using a weak axiom of countable choice, one may also prove ${\textbf {LLPO}} \rightarrow {\textbf {WKL}}$, see Theorem 4.3.

2.2.15 The limited principle of omniscience, ${\textbf {LPO}}$:

$$\begin{aligned} \forall \alpha [\exists n[\alpha (n) \ne 0] \;\vee \; \forall n[\alpha (n) = 0]]. \end{aligned}$$

${\textbf {LPO}}$ and ${\textbf {LLPO}}$ were introduced by E. Bishop, see [3, Chapter 1, Sect. 1]. The following result is not difficult and well-known, see [21, Section 2.6] and [1, Theorem 3.1].

Theorem 2.11

$\textsf{BIM} \vdash \textbf{LPO} \rightarrow \textbf{LLPO}$.

Proof

Let $\alpha$ be given. Apply ${\textbf {LPO}}$ and distinguish two cases.

Case (1). $\exists n[\alpha (n) \ne 0]$. Find q, k such that $k<2$ and $2q+k=\mu n[\alpha (n)\ne 0]$. Conclude: $\forall p[2p+1-k\ne \mu n[\alpha (n)\ne 0]]$.

Case (2). $\forall n[\alpha (n) = 0]$. Then $\forall k<2\forall p[2p+k\ne \mu n[\alpha (n)\ne 0]]$. $\square$

The following Lemma shows that, in $\textsf{BIM}+{\textbf {BCP}}$, not every closed subset of $\omega ^\omega$ is a spread, see also [47, Theorem 2.10 (vi)].

Lemma 2.12

$\textsf{BIM}\vdash \forall \beta \exists \gamma [Spr(\gamma )\;\wedge \; \mathcal {F}_\beta =\mathcal {F}_\gamma ]\rightarrow \textbf{LPO}$.

Proof

Assume $\forall \beta \exists \gamma [Spr(\gamma )\;\wedge \; \mathcal {F}_\beta =\mathcal {F}_\gamma ]$. We will prove ${\textbf {LPO}}$. Let $\alpha$ be given. Define $\beta$ such that $\beta (0)=0$ and $\forall n \forall s[\beta (\langle n \rangle *s)=0 \leftrightarrow \alpha (n)\ne 0]$. Assume we find $\gamma$ such that $Spr(\gamma )$ and $\mathcal {F}_\beta =\mathcal {F}_\gamma$. If $\gamma (0)=0$, then $\exists n[\alpha (n)\ne 0]$ and, if $\gamma (0)\ne 0$, then $\forall n[\alpha (n)=0]$. We thus see $\forall \alpha [\exists n[\alpha (n)\ne 0]\;\vee \; \forall n[\alpha (n)=0]]$, i.e. ${\textbf {LPO}}$.$\square$

2.2.16 Markov’s principle, ${\textbf {MP}}$:

$$\begin{aligned} \forall \alpha [\lnot \lnot \exists n[\alpha (n)\ne 0]\rightarrow \exists n[\alpha (n)\ne 0]]. \end{aligned}$$

For some discussion of this principle, see [30, Volume I, Chapter 4, Section 5]. In this paper, the principle figures only in Sect. 12.1.

We would like to make a philosophical observation. For a constructive mathematician, the assumptions ${\textbf {LPO}}$, ${\textbf {WKL}}$, ${\textbf {LLPO}}$ and ${\textbf {MP}}$ make no sense, as she does not know a situation in which these assumptions are true. Theorem 2.10: ${\textbf {WKL}} \rightarrow {\textbf {LLPO}}$ concludes something which is never true from something which is never true. Nevertheless, the proof of Theorem 2.10 makes sense. It shows us how to find, given any $\alpha$, a suitable $\beta$ such that if $\beta$ has the ${\textbf {WKL}}$-property, then $\alpha$ has the ${\textbf {LLPO}}$-property. This part of the argument does not use the assumption that every $\beta$ has the ${\textbf {WKL}}$-property. Theorems 2.6 and 2.11 deserve a similar comment.

The reader may find more information on the axioms of intuitionistic analysis in [5, 6, 8, 9, 13, 30] and [46].

3 The $\varvec{\Sigma }^0_1$-separation principle

3.1. In classical reverse mathematics, weak König’s Lemma is equivalent to a principle called $\Sigma ^0_1$-separation, see [27, Lemma IV.4.4]. We call $X\subseteq \omega$ enumerable or $\varvec{\Sigma }^0_1$ if and only if $\exists \alpha [X= E_\alpha ]$.The following statement, formulated in the intuitionistic language of $\textsf{BIM}$, comes close to the just-mentioned classical principle.

$\varvec{\Sigma }^0_1$-separation principle, $\varvec{\Sigma }_1^0$-${\textbf {Sep}}$:

$$\begin{aligned} \forall \alpha [\lnot \exists n\forall i<2[(n,i)\in E_\alpha ]\rightarrow \exists \gamma \forall n[ \bigl (n,\gamma (n)\bigr )\notin E_\alpha ]]. \end{aligned}$$

The next Theorem seems to confirm the just-mentioned classical result.

Theorem 3.1

$\textsf{BIM}\vdash \textbf{WKL} \leftrightarrow \varvec{\Sigma }^0_1$-$\textbf{Sep}$.

Proof

(i) Assume ${\textbf {WKL}}$. We will prove $\varvec{\Sigma }^0_1$-${\textbf {Sep}}$.

Let $\alpha$ be given such that $\lnot \exists n\forall i<2[(n,i)\in E_\alpha ]$. Define $\beta$ such that $\forall a\in 2^{<\omega }[\beta (a) = 0\leftrightarrow \forall m <length(a)[\bigl (m,a(m)\bigr )\notin E_{{\overline{\alpha }} n} ]]$. Let n be given. Note that $\forall m<n\exists i<2[(m,i)\notin E_{{\overline{\alpha }} n}]$ and find a in $2^{<\omega }$ such that $length(a)=n$ and $\forall m<n[\bigl (m,a(m)\bigr )\notin E_{{\overline{\alpha }} n}]$. Conclude that $\forall j \le m [\beta ({\overline{a}} j) = 0]$. We thus see that $\forall n\exists a\in 2^{<\omega }[length(a)=n \;\wedge \;\forall m\le n[\beta ({\overline{a}} n)=0]]$. Using ${\textbf {WKL}}$, find $\gamma$ in $2^\omega$ such that $\forall n[\beta ({\overline{\gamma }} n) = 0]$. Conclude that $\forall n \forall m<n[\bigl (m, \gamma (m)\bigr )\notin E_{{\overline{\alpha }} n}]$ and $\forall n[\bigl (n,\gamma (n)\bigr ) \notin E_{\alpha }]$. We thus see that $\forall \alpha [\lnot \exists n\forall i<2[(n,i)\in E_\alpha ]\rightarrow \exists \gamma \forall n[\bigl (n, \gamma (n)\bigr )\notin E_\alpha ]$, i.e. $\varvec{\Sigma }^0_1$-${\textbf {Sep}}$.

(ii) Assume $\varvec{\Sigma }^0_1$-${\textbf {Sep}}$. We will prove ${\textbf {WKL}}$.

Let $\alpha$ be given such that $\forall m [\lnot Bar_{2^\omega }(D_{{\overline{\alpha }} m})]$. Define $\beta$ such that, for both $i<2$, for all n, for all s, if $s\in 2^{<\omega }$ and $Bar_{2^\omega \cap s*\langle i \rangle }(D_{{\overline{\alpha }} n})$ and $\lnot Bar_{2^\omega \cap s *\langle 1-i \rangle } (D_{{\overline{\alpha }} n})$, then $\beta (n,s) = (s,i)+1$, and, if not, then $\beta (n,s) =0$. Note that $E_\beta$ is the set of all (s, i) such that $s \in 2^{<\omega }$ and $i<2$ and $\exists n[Bar_{2^\omega \cap s*\langle i \rangle }(D_{{\overline{\alpha }} n})\;\wedge \;\lnot Bar_{2^\omega \cap s*\langle 1-i \rangle }(D_{{\overline{\alpha }} n})]$. Note that, for all s in $2^{<\omega }$, for all $i<2$, if $(s,i)\notin E_\beta$, then $\forall n[Bar_{2^\omega \cap s*\langle i \rangle }(D_{{\overline{\alpha }} n})\rightarrow Bar_{2^\omega \cap s*\langle 1-i \rangle }(D_{{\overline{\alpha }} n})]$. Note that $\lnot \exists s\forall i <2[(s,i)\in E_\beta ]$. Using $\varvec{\Sigma }^0_1$-${\textbf {Sep}}$, find $\gamma$ in $2^\omega$ such that $\forall s[\bigl (s,\gamma (s)\bigr ) \notin E_{\beta }]$. Define $\delta$ in $2^\omega$ such that $\forall n[\delta (n)= \gamma (\overline{\delta }n)]$.

Suppose we find n such that $\alpha (\overline{\delta }n) \ne 0$. Define $q:={\overline{\delta }} n +1$ and note that ${\overline{\delta }} n \in D_{{\overline{\alpha }} q}$, i.e. $Bar_{2^\omega \cap {\overline{\delta }} n}(D_{{\overline{\alpha }} q})$. We prove, using backwards induction, that $\forall j\le n[Bar_{2^\omega \cap {\overline{\delta }} j}(D_{{\overline{\alpha }} q})]$. Observe that $Bar_{2^\omega \cap {\overline{\delta }} n}(D_{{\overline{\alpha }} q})$. Now suppose $j+1 \le n$ and $Bar_{2^\omega \cap {\overline{\delta }} (j+1)}(D_{{\overline{\alpha }} q})$. Note that ${\overline{\delta }} (j+1)={\overline{\delta }} j*\langle \gamma ({\overline{\delta }} j)\rangle$. Also $\bigl ({\overline{\delta }} j, \gamma ({\overline{\delta }} j)\bigr )\notin E_\beta$. Conclude that $Bar_{2^\omega \cap {\overline{\delta }} j*\langle 1-\gamma ({\overline{\delta }}(j)\rangle } (D_{{\overline{\alpha }} q})$ and $Bar_{2^\omega \cap {\overline{\delta }} j} (D_{{\overline{\alpha }} q})$. This completes the proof of the induction step. After n steps we find that $Bar_{2^\omega }(D_{{\overline{\alpha }} q})$. This contradicts the assumption $\forall m[\lnot Bar_{2^\omega }(D_{{\overline{\alpha }} m})]$. Conclude that $\forall n[\alpha (\overline{\delta }n) = 0]$.

We thus see that $\forall \alpha [\forall n[\lnot Bar_{2^\omega }(D_{{\overline{\alpha }} n})]\rightarrow \exists \delta \in 2^\omega \forall n[\alpha ({\overline{\delta }} n)=0]]$, i.e. ${\textbf {WKL}}$. $\square$

Theorem 3.1 shows that $\varvec{\Sigma }^0_1$-${\textbf {Sep}}$, like ${\textbf {WKL}}$, is not constructive, see Theorem 2.10. It is not true in intuitionistic analysis and it also fails in the model of $\textsf{BIM}$ given by the recursive functions.

4 ${\textbf {AC}}_{\omega ,\omega }$, some special cases

The following restricted version of ${\textbf {AC}}_{\omega ,\omega }$ is provable in $\textsf{BIM}$ as it is a consequence of Axiom 3, see Sect. 2.

4.1. Minimal Axiom of Countable Choice, $\varvec{\Delta }^0_1$-${\textbf {AC}}_{\omega ,\omega }$:

$$\begin{aligned}\forall \alpha [ \forall n \exists m [ \alpha (n,m) = 0 ] \rightarrow \exists \gamma \forall n [ \alpha \bigl (n, \gamma (n)\bigr ) = 0 ] ].\end{aligned}$$

4.2. Axiom Scheme of Countable Unique Choice, ${\textbf {AC}}_{\omega ,\omega }!={\textbf {AC}}_{0,0}!$:

$$\begin{aligned}\forall n\exists !m[R(n,m)]\rightarrow \exists \gamma \forall n[R\bigl (n,\gamma (n)\bigr )].\end{aligned}$$

where ‘$\forall n\exists !m[R(n,m)]$’ abbreviates ‘$\forall n\exists m[R(n,m)\;\wedge \;\forall p[R(n,p)\rightarrow m=p]]$’.

${\textbf {AC}}_{\omega ,\omega }!$ is not a theorem of $\textsf{BIM}$, see Sects. 4.8 and [31].

4.3. $\varvec{\Sigma }^0_1$-First Axiom of Countable Choice, $\varvec{\Sigma }^0_1$-${\textbf {AC}}_{\omega ,\omega }$:

$$\begin{aligned}\forall \alpha [\forall n \exists m[(n,m) \in E_{\alpha }] \rightarrow \exists \gamma \forall n[\bigl (n,\gamma (n)\bigr ) \in E_{\alpha }]].\end{aligned}$$

Theorem 4.1

$\textsf{BIM}\vdash \varvec{\Sigma }^0_1$-AC$_{\omega ,\omega }$.

Proof

Assume $\forall n \exists m[(n,m) \in E_{\alpha }]$. Then $\forall n \exists m \exists p[\alpha (p) = (n,m)+1]$. Find $\delta$ such that $\forall n[\delta (n)=\mu q[\alpha (q')=(n, q'')+1]]$. Define $\gamma$ such that $\forall n[\gamma (n) = \delta ''(n)]$ and note that $\forall n[\bigl (n,\gamma (n)\bigr ) \in E_{\alpha }]$.$\square$

We call $X\subseteq \omega$ co-enumerable or $\varvec{\Pi }^0_1$ if and only if there exists $\alpha$ such that $X= \omega \setminus E_\alpha$.

4.4. $\varvec{\Pi }_1^0$-First Axiom of Countable Choice, $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$:

$$\begin{aligned}\forall \alpha [ \forall n \exists m[(n, m) \notin E_{\alpha }] \rightarrow \exists \gamma \forall n [\bigl (n,\gamma (n)\bigr ) \notin E_{\alpha }]].\end{aligned}$$

$\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$ is unprovable in $\textsf{BIM}$, see Sect. 4.8. In [44, Section 6], we introduced the following special case of this axiom.

4.5.Weak $\varvec{\Pi }^0_1$-First Axiom of Countable Choice, Weak-$\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$:

$$\begin{aligned}\forall \alpha [\forall m \exists n \forall p \ge n[\alpha (m,p) \ne 0] \rightarrow \exists \gamma \forall m \forall p \ge \gamma (m)[\alpha (m,p) \ne 0]].\end{aligned}$$

Weak-$\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$ follows from ${\textbf {AC}}_{\omega ,\omega }!={\textbf {AC}}_{0,0}!$. Weak-$\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$ is a special case of the axiom scheme $AC_{0,0}^m$ introduced in [23, Sect. 3.1]. We suspect that, in $\textsf{BIM}$, Weak-$\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$ does not imply $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$, but we have no proof.¹⁰

Theorem 4.2

• (i) $\textsf{BIM}+$Weak-$\varvec{\Pi }^0_1$-AC$_{\omega ,\omega }\vdash \forall \beta [Fan(\beta )\rightarrow Fan^+(\beta )]$.

• (ii) $\textsf{BIM}+$Weak-$\varvec{\Pi }^0_1$-AC$_{\omega ,\omega }\vdash \textbf{FT}\rightarrow \textbf{FT}^+$.

Proof

The proof is left to the reader. $\square$

One may also study statements one obtains from ${\textbf {AC}}_{\omega ,\omega }$ by limiting the number of alternatives one has at each choice.

4.6. Axiom Scheme of Countable Binary Choice, ${\textbf {AC}}_{\omega ,2}$:

$$\begin{aligned} \forall n\exists m<2[R(n,m)] \rightarrow \exists \gamma \in 2^\omega \forall n[R\bigl (n,\gamma (n)\bigr )]. \end{aligned}$$

Here is a restricted version of ${\textbf {AC}}_{\omega ,2}$:

4.7. $\varvec{\Pi }^0_1$-Axiom of Countable Binary Choice, $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$:

$$\begin{aligned}\forall \alpha [\forall n\exists m <2[(n,m) \notin E_{\alpha } ] \rightarrow \exists \gamma \in 2^\omega \forall n[\bigl (n,\gamma (n)\bigr ) \notin E_{\alpha }]].\end{aligned}$$

A result related to the following Theorem has been proven by U. Kohlenbach, see [16, Theorem 3]. A similar result is mentioned in [1, Sect. 2.2].

Theorem 4.3

$\textsf{BIM}\vdash (\varvec{\Pi }^0_1$-$\textbf{AC}_{\omega ,2}\;\wedge \; \textbf{LLPO}) \leftrightarrow \textbf{WKL}$.

Proof

(i) Assume, in $\textsf{BIM}$, $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ and ${\textbf {LLPO}}$. It suffices to prove $\varvec{\Sigma }^0_1$-${\textbf {Sep}}$, as, according to Theorem 3.1, $\textsf{BIM}\vdash \varvec{\Sigma }^0_1$-${\textbf {Sep}}\leftrightarrow {\textbf {WKL}}$.

Let $\alpha$ be given such that $\forall n\lnot \forall m<2[(n,m)\in E_\alpha ]$. Let n be given. Define $\beta$ such that $\forall q\forall i<2[\beta (2q+i) \ne 0\leftrightarrow q=\mu p[\alpha (p)=(n,i)+1]]$. Apply ${\textbf {LLPO}}$ and find $i<2$ such that $\forall q[2q+i\ne \mu m[\beta (m)\ne 0]]$. Assume $(n,i)\in E_\alpha$. Then $(n, 1-i)\notin E_\alpha$ and $\lnot \exists p[\alpha (p)=(n, 1-i)+1]$ and $\forall q[\beta (2q+1-i)=0]$. Find $q:=\mu p[\alpha (p)= (n,i)+1]$. Note $\beta (2q+i)\ne 0$ and $\forall m<2q+i[\beta (m)=0]$, so $2q+i=\mu m[\beta (m)\ne 0]$. Contradiction. Conclude that $(n,i)\notin E_\alpha$. Conclude that $\forall n\exists i<2[(n,i)\notin E_\alpha ]$. Apply $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ and find $\gamma$ in $2^\omega$ such that $\forall n[\bigl (n,\gamma (n)\bigr )\notin E_\alpha ]$. Conclude that $\forall \alpha [\forall n[\lnot \forall m<2[(n,m)\in E_\alpha ]\rightarrow \exists \gamma \in 2^\omega \forall n[\bigl (n,\gamma (n)\bigr ))\notin E_\alpha ]]$, i.e. $\varvec{\Sigma }^0_1$-${\textbf {Sep}}$.

(ii) Note that $\lnot (P\;\wedge \;Q)\leftrightarrow \lnot \lnot (\lnot P\;\vee \;\lnot Q)$ is a valid scheme of intuitionistic logic. Conclude that $\textsf{BIM}\vdash \varvec{\Sigma }^0_1$-${\textbf {Sep}}\leftrightarrow \bigl (\forall \alpha [\forall n\lnot \lnot \exists m < 2[(n,m) \notin E_{\alpha } ] \rightarrow \exists \gamma \in 2^\omega \forall n[ \bigl (n,\gamma (n)\bigr ) \notin E_{\alpha }]]\bigr ).$ Conclude that $\textsf{BIM}\vdash \varvec{\Sigma }^0_1$-${\textbf {Sep}}\rightarrow \varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, and, using Theorem 3.1, that $\textsf{BIM}\vdash {\textbf {WKL}}\rightarrow \varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$. The conclusion $\textsf{BIM}\vdash {\textbf {WKL}}\rightarrow {\textbf {LLPO}}$ has been drawn in Theorem 2.10. $\square$

From a constructive point of view, $\varvec{\Sigma }^0_1$-${\textbf {Sep}}$, or equivalently, ${\textbf {WKL}}$, is an axiom of countable choice that is formulated too strongly.

4.1 ${\textbf {AC}}_{\omega ,2}!$ and $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ are unprovable in $\textsf{BIM}$

Note that the theory $\textsf{BIM} +{\textbf {CT}}$ may be translated into intuitionistic arithmetic $\textsf{HA}$, by interpreting functions from $\omega$ to $\omega$ as indices of total computable functions. The negative translation due to Gödel and Gentzen, see [30, vol. I, Ch. 3, Subsection 3.4], shows that first order classical (Peano) arithmetic $\textsf{PA}$, the theory that results from $\textsf{HA}$ by adding the axiom scheme $X\;\vee \lnot X$, is consistent. It follows that also the theory $\textsf{BIM}+{\textbf {CT}}$ remains consistent upon adding the axiom scheme $X \;\vee \;\lnot X$.

4.8.1. Note that the theory $\textsf{BIM}+{\textbf {CT}}+\; X\vee \lnot X\;+{\textbf {AC}}_{\omega ,2}!$ is inconsistent. The argument is as follows.

Let $\tau , \psi$ be as in ${\textbf {CT}}$. Define $H:=\{n\mid \exists z[\tau (n,n,z)\ne 0]\}$. Using classical logic, conclude: $\forall n\exists ! i<2[i=0\leftrightarrow n\in H]$. Using ${\textbf {AC}}_{\omega , 2}!$, find $\alpha$ such that $\forall n[\alpha (n)\ne 0\leftrightarrow n \in H]$. Define $\beta$ such that, for each n, if $\alpha (n)\ne 0$, then $\beta (n) = \psi (\mu z[\tau (n,n,z)\ne 0])+1$, and, if $\alpha (n)=0$, then $\beta (n)=0$. Using ${\textbf {CT}}$, find $n_0$ such that $\forall n[\beta (n)=\psi (\mu z[\tau (n_0, n,z)\ne 0])]$. Note that $\alpha (n_0)\ne 0$ and: $\beta (n_0)=\beta (n_0)+1$. Contradiction.

Conclude that, if $\textsf{HA}$ is consistent, then ${\textbf {AC}}_{\omega , 2}!$ is not derivable in $\textsf{BIM}$.

4.8.2. Theorem 4.3 implies that $\textsf{BIM} + \lnot !{\textbf {FT}} + {\textbf {LLPO}} + \varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ is not consistent, as $\lnot !{\textbf {FT}}$ contradicts ${\textbf {WKL}}$. Conclude that $\textsf{BIM} + \lnot !{\textbf {FT}} + {\textbf {LLPO}}\vdash \lnot \varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$.

On the other hand, $\textsf{BIM} + \lnot !{\textbf {FT}} + {\textbf {LLPO}}$ is consistent, as it is a subsystem of $\textsf{BIM} + {\textbf {CT}}+ \;X\vee \lnot X$. It follows that $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ is not derivable in $\textsf{BIM}+\lnot !{\textbf {FT}}+{\textbf {LLPO}}$ and, a fortiori, not derivable in $\textsf{BIM}$. The stronger axiom $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$ and the even stronger axiom $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega ^\omega }$, to be introduced in Sect. 6, also are not derivable in $\textsf{BIM}$.

From Theorem 4.3, we also conclude that $\textsf{BIM} + \lnot !{\textbf {FT}}+ \varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}\vdash \lnot {\textbf {LLPO}}$.

In the context of $\textsf{HA}$, Church’s Thesis is sometimes introduced as an axiom scheme, $\textsf{CT}_0$, see [30, vol. I, Ch. 4, Sect. 3]:

$$\begin{aligned} \forall n \exists m[A(n,m)] \rightarrow \exists e \forall n \exists z[T(\langle e,n,z\rangle ) \;\wedge \; \forall i<z[\lnot T(\langle e, n, i \rangle )] \;\wedge \; A\bigl (n,(U(z)\bigr )]. \end{aligned}$$

It is not difficult to see that $\textsf{BIM}+ {\textbf {CT}}+{\textbf {AC}}_{\omega ,\omega }$ and also $\textsf{BIM}+ {\textbf {CT}}+{\textbf {AC}}_{\omega ,\omega ^\omega }$ translate into $\textsf{HA}+\textsf{CT}_0$.¹¹ There is no straightforward model for $\textsf{HA} +\mathsf {CT_0}$ but, using realizability, one may show that, if $\textsf{HA}$ is consistent, then so is $\textsf{HA}+\mathsf {CT_0}$, see [30, vol. I, Ch. 4, Sect. 4]. It follows that, if $\textsf{HA}$ is consistent, then $\textsf{BIM}+\lnot !{\textbf {FT}}+\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ is consistent.

The following axiom scheme may be compared to the axiom $\text {BC}_{0,0}$ of bounded countable choice occurring in [23, Section 3.2].

4.9. Axiom Scheme of Countable Finite Choice, ${\textbf {AC}}_{\omega ,<\omega }$:

$$\begin{aligned} \forall \beta [\forall n\exists m \le \beta (n)[ R(n,m)] \rightarrow \exists \gamma \forall n[\gamma (n) \le \beta (n) \;\wedge \; R\bigl (n,\gamma (n)\bigr )]]. \end{aligned}$$

4.10. $\varvec{\Pi }^0_1$-Axiom of Countable Finite Choice, $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,<\omega }$:

$$\begin{aligned} \forall \alpha [\forall n\exists m \le \alpha ^{\upharpoonright 0}(n)[(n,m) \notin E_{\alpha ^{\upharpoonright 1}}] \rightarrow \exists \gamma \forall n[\gamma (n)\le \alpha ^{\upharpoonright 0}(n)\;\wedge \;\bigl (n,\gamma (n)\bigr ) \notin E_{\alpha ^{\upharpoonright 1}}]].\end{aligned}$$

We conjecture that $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,<\omega }$ is not provable in $\textsf{BIM}+ \varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, but we have no proof of this conjecture. There might be many statements intermediate in strength like $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,3}$, the $\varvec{\Pi }^0_1$-Axiom of Countable Ternary Choice.

5 Contrapositions of some special cases of ${\textbf {AC}}_{\omega ,\omega }$

The following statement is a contraposition of ${\textbf {AC}}_{\omega ,\omega }$:

5.1. First axiom scheme of reverse countable choice, $\overleftarrow{{\textbf {AC}}_{\omega ,\omega }}$:

$$\begin{aligned} \forall \gamma \exists n[R\bigl (n,\gamma (n)\bigr )] \rightarrow \exists n \forall m[R(n,m)]. \end{aligned}$$

A special case is:

5.2. Minimal axiom of reverse countable choice, $\varvec{\Delta }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,\omega }}:$

$$\begin{aligned} \forall \alpha [\forall \gamma \exists n[\bigl (n, \gamma (n)\bigr )\in D_\alpha ] \rightarrow \exists n \forall m[(n,m)\in D_\alpha ]]. \end{aligned}$$

The following result may be found in [33, Section 2].

Theorem 5.1

$\textsf{BIM}+ \varvec{\Delta }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,\omega }}\vdash \textbf{LPO}$.

Proof

Assume $\varvec{\Delta }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,\omega }}$. We will prove ${\textbf {LPO}}$.

Let $\beta$ be given. Define $\alpha$ such that $\forall n\forall m[\alpha (n, m) = 0\leftrightarrow ({\overline{\beta }} n= \underline{{\overline{0}}}n\;\wedge \; {\overline{\beta }} m\ne \underline{{\overline{0}}}m)]$. Note that, for every $\gamma$, either ${\overline{\beta }} \bigl (\gamma (0)\bigr )= \overline{\underline{0}}\bigl (\gamma (0)\bigr )$ and $\alpha \bigl (0, \gamma (0)\bigr ) \ne 0$, i.e. $\bigl (0, \gamma (0)\bigr )\in D_\alpha$, or ${\overline{\beta }} \bigl (\gamma (0)\bigr )\ne \overline{\underline{0}}\bigl (\gamma (0)\bigr )$, and $\alpha \bigl (\gamma (0), \gamma (\gamma (0))\bigr )\ne 0$, i.e. $\bigl (\gamma (0), \gamma (\gamma (0))\bigr )\in D_\alpha$. Conclude that $\forall \gamma \exists n[\bigl (n, \gamma (n)\bigr ) \in D_\alpha ]$. Applying $\varvec{\Delta }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,\omega }}$, find n such that $\forall m[(n,m) \in D_\alpha ]$. Either ${\overline{\beta }} n \ne \overline{\underline{0}}n$ and $\exists j [\beta (j) \ne 0]$, or ${\overline{\beta }} n = \overline{\underline{0}}n$. In the latter case, for each m, ${\overline{\beta }} m = \overline{\underline{0}}m$ and $\forall j[\beta (j) = 0]$. We thus see $\forall \beta [\exists n[\beta (n)\ne 0]\;\vee \;\forall n[\beta (n)=0]]$, i.e. ${\textbf {LPO}}$. $\square$

5.3. Axiom scheme of reverse countable binary choice, $\overleftarrow{{\textbf {AC}}_{\omega ,2}}:$

$$\begin{aligned} \forall \gamma \in 2^\omega \exists n[R\bigl (n,\gamma (n)\bigr )] \rightarrow \exists n\forall i<2[R(n,i)]. \end{aligned}$$

In [33, Section 4], $\overleftarrow{{\textbf {AC}}_{\omega ,2}}$ has been shown to be a consequence of ${\textbf {FT}}+{\textbf {AC}}_{\omega , \omega ^\omega }$.

We introduce a restricted version:

$\varvec{\Delta }^0_1$-Axiom of Reverse Countable Binary Choice, $\varvec{\Delta }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}}$:

$$\begin{aligned}\forall \alpha [ \forall \gamma \in 2^\omega \exists n[\bigl (n,\gamma (n)\bigr ) \in D_{\alpha }] \rightarrow \exists n\forall i<2[ (n,i) \in D_{\alpha }]].\end{aligned}$$

Theorem 5.2

$\textsf{BIM}\vdash \varvec{\Delta }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}}$.

Proof

Let $\alpha$ be given such that $\forall \gamma \in 2^\omega \exists n[\bigl (n,\gamma (n)\bigr ) \in D_{\alpha }]$. Define $\gamma$ in $2^\omega$ such that $\forall n[(n,0)\in D_\alpha \leftrightarrow \gamma (n) = 1]$. Find n such that $\bigl (n,\gamma (n)\bigr )\in D_\alpha$. Note that $\gamma (n) = 1$ and $\forall i<2[(n,i) \in D_\alpha ]$. $\square$

We now introduce a less restricted version:

5.4. $\varvec{\Sigma }_1^0$-Axiom of reverse countable binary choice, $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}}$:

$$\begin{aligned}\forall \alpha [ \forall \gamma \in 2^\omega \exists n[\bigl (n,\gamma (n)\bigr ) \in E_{\alpha }] \rightarrow \exists n\forall i<2[ (n,i) \in E_{\alpha }]].\end{aligned}$$

We define a formula that we want to call its strong negation:

5.5. $\lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}})$:

$$\begin{aligned} \exists \alpha [ \forall \gamma \in 2^\omega \exists n[\bigl (n,\gamma (n)\bigr ) \in E_{\alpha }] \;\wedge \; \lnot \exists n\forall i<2[(n, i) \in E_\alpha ] ]. \end{aligned}$$

Note that $\textsf{BIM}$ proves $\forall \alpha [\lnot \exists n\forall i<2[(n, i) \in E_\alpha ]\leftrightarrow \forall n\forall p\forall q[\alpha (p) = (n,0)+1\rightarrow \alpha (q)\ne (n,1)+1]]$.

Lemma 5.3

$\textsf{BIM}$ proves the following:

• (i) $\textbf{FT} \rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf{AC}}_{\omega ,2}}$ and $\lnot !({\varvec{\Sigma }}^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}}) \rightarrow \lnot !\textbf{FT}$.

• (ii) $\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}}\rightarrow \textbf{FT}$ and $\lnot !\textbf{FT} \rightarrow \lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}})$.

Proof

(i) We prove, in $\textsf{BIM}$, that, for each $\alpha$, there exists $\beta$ such that

$$\begin{aligned}{} & {} (1)\;\forall \gamma \in 2^\omega \exists n[\bigl (n,\gamma (n)\bigr ) \in E_{\alpha }] \rightarrow Bar_{2^\omega }(D_\beta )\;\text {and}\\{} & {} (2)\;\exists m[Bar_{2^\omega }(D_{{\overline{\beta }} m})] \rightarrow \exists n\forall i<2[(n,i)\in E_\alpha ]. \end{aligned}$$

The two promised conclusions then follow easily.

Let $\alpha$ be given. Define $\beta$ such that

$$\begin{aligned} \forall a \in 2^{<\omega }[\beta (a) \ne 0\leftrightarrow \exists n<length(a)[\bigl (n, a(n)\bigr )\in E_{{\overline{\alpha }} length(a)}]]. \end{aligned}$$

(1) Assume $\forall \gamma \in 2^\omega \exists n [ \bigl (n,\gamma (n)\bigr ) \in E_{\alpha }]$. Let $\gamma$ in $2^\omega$ be given. Find n, p such that $\bigl (n,\gamma (n)\bigr )\in E_{{\overline{\alpha }} p}$ and $n<p$. Note that $\beta ({\overline{\gamma }} p) \ne 0$. We thus see that $\forall \gamma \in 2^\omega \exists p[\beta ({\overline{\gamma }} p)\ne 0]$, i.e. $Bar_{2^\omega }(D_\beta )$.

(2) Let m be given such that $Bar_{2^\omega }(D_{{\overline{\beta }} m})$. Note that $\forall a \in 2^{<\omega }[length(a) <a]$ and $\forall \gamma \in 2^\omega \exists n<m [\beta ({\overline{\gamma }} n)\ne 0 ]$. It follows that $\forall a\in 2^{<\omega }[length(a)=m\rightarrow \exists n<m[\beta ({\overline{a}} n)\ne 0]]$. Assume $\forall n<m\exists i<2[ (n,i)\notin E_{{\overline{\alpha }} m}]$. Define a in $2^{<\omega }$ such that $length(a)=m$ and $\forall n<m[(n,0)\notin E_{{\overline{\alpha }} m}\leftrightarrow a(n)=0]$. Conclude that $\forall n<m[\bigl (n, a(n)\bigr )\notin E_{{\overline{\alpha }} m}]$ and $\forall n \le m m[\beta ({\overline{a}} n)= 0]$. Contradiction. Conclude that $\exists n<m\forall i<2[(n,i)\in E_{{\overline{\alpha }} m}\subseteq E_\alpha ]$.

(ii)¹² We prove, in $\textsf{BIM}$: for each $\alpha$, there exists $\beta$ such that

$$\begin{aligned}{} & {} (1)\;Bar_{2^\omega }(D_\alpha )\rightarrow \forall \gamma \in 2^\omega \exists n[\bigl ( n, \gamma (n) \bigr ) \in E_{\beta }]\;\text {and}\\{} & {} (2)\;\exists n\forall i<2[(n,i)\in E_\beta ]\rightarrow \exists m[Bar_{2^\omega }(D_{{\overline{\alpha }} m})]. \end{aligned}$$

The two promised conclusions then follow easily.

Let $\alpha$ be given. Define $\beta$ such that, for all n, for every s in $2^{<\omega }$, for all $i<2$, if either (a) $Bar_{2^\omega }(D_{{\overline{\alpha }} n})$, or (b) $Bar_{2^\omega \cap s*\langle i \rangle }(D_{{\overline{\alpha }} n})$ and not $Bar_{2^\omega \cap s*\langle 1- i \rangle }(D_{{\overline{\alpha }} n})$, then $\beta (n, s*\langle i \rangle ) = (s,i)+1$, and, (c) if both (a) and (b) fail, then $\beta (n,s*\langle i \rangle )=0$. Furthermore, for all n, for all s, if $s\notin 2^{<\omega }$, then $\beta (n,s)=0$. Note that $E_\beta$ is the set of all pairs (s, i) such that $s\in 2^{<\omega }$ and $i<2$ and either $\exists n[Bar_{2^\omega }(D_{{\overline{\alpha }} n})]$ or $\exists n[Bar_{2^\omega \cap s*\langle i\rangle }(D_{{\overline{\alpha }} n})\;\wedge \lnot Bar_{2^\omega \cap s*\langle 1- i\rangle }(D_{{\overline{\alpha }} n})]$. Note that, for all s in Bin, if $\forall i<2[(s,i) \in E_\beta ]$, then $\exists n[Bar_2^\omega (D_{{\overline{\alpha }} n})]$.

(1) Assume $Bar_{2^\omega }(D_\alpha )$. We will prove that $\forall \gamma \in 2^\omega \exists n[\bigl (n,\gamma (n)\bigr )\in E_\beta ]$. Let $\gamma$ in $2^\omega$ be given. Define $\delta$ in $2^\omega$ such that $\forall n [\delta (n) = \gamma (\overline{\delta }n)]$. Find n such that $\alpha (\overline{\delta }n) \ne 0$. Define $q:={\overline{\delta }} n +1$ and note: ${\overline{\delta }} n\in D_{{\overline{\alpha }} q}$. We claim that for all $j \le n$, either $\exists i \le n[\bigl ({\overline{\delta }} i, \gamma ({\overline{\delta }} i)\bigr ) \in E_{\beta }]$, or $Bar_{2^\omega \cap {\overline{\delta }} j}(D_{{\overline{\alpha }} q})$. We prove this claim by backwards induction, starting from $j = n$. Note that $Bar_{2^\omega \cap \overline{\delta }n}(D_{{\overline{\alpha }} q})$. Now assume $j < n$ and $Bar_{2^\omega \cap {\overline{\delta }}(j+1)}(D_{{\overline{\alpha }} q})$, i.e. $Bar_{2^\omega \cap \overline{\delta }(j)*\langle \delta (j) \rangle }(D_{{\overline{\alpha }} q})$. Find out if also $Bar_{2^\omega \cap \overline{\delta }(j)*\langle 1-\delta (j) \rangle }(D_{{\overline{\alpha }} q})$. If so, then $Bar_{2^\omega \cap \overline{\delta }(j)}(D_{{\overline{\alpha }} q})$, and, if not, then $\beta \bigl (q,\overline{\delta }(j+1)\bigr )= \bigl ({\overline{\delta }} j, \delta (j)\bigr ) + 1$, and $\bigl ({\overline{\delta }} j, \delta (j)\bigr )=\bigl ({\overline{\delta }} j, \gamma ({\overline{\delta }} (j)\bigr )\in E_\beta$. We may conclude that either $\exists j\le n[({\overline{\delta }} j, \delta (j)\bigr )\in E_\beta ]$ or $Bar_{2^\omega }(D_{{\overline{\alpha }} q})$. Note that, if $Bar_{2^\omega }(D_{{\overline{\alpha }} q})$, then $\forall s \in 2^{<\omega }\forall i<2[\beta (q,s*\langle i \rangle ) = (s, i) +1]$ and $\forall s\in 2^{<\omega }[\bigl (s, \gamma (s)\bigr ) \in E_\beta ]$. We thus see that $\forall \gamma \in 2^\omega \exists s[ \bigl (s,\gamma (s)\bigr ) \in E_{\beta }]$.

(2) Assume that $\exists n\forall i<2[(n,i)\in E_\beta ]$. Conclude, using the observation we made just after the definition of $\beta$, that $\exists n[Bar_{2^\omega }(D_{{\overline{\alpha }} n})]$. $\square$

Theorem 5.4

$\textsf{BIM}$ proves: $\textbf{FT} \leftrightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}}$ and: $\lnot !\textbf{FT} \leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}})$.

Proof

Use Lemma 5.3.$\square$

5.6. Axiom Scheme of Reverse Countable Finite Choice, $\overleftarrow{{\textbf {AC}}_{\omega ,<\omega }}$:

$$\begin{aligned} \forall \beta [ \forall \gamma \exists n[\gamma (n)\le \beta (n) \rightarrow R\bigl (n,\gamma (n)\bigr )]\rightarrow \exists n \forall m \le \beta (n)[R(n,m) ]]. \end{aligned}$$

$\overleftarrow{{\textbf {AC}}_{\omega ,<\omega }}$ may be concluded from¹³${\textbf {FT}}+{\textbf {AC}}_{\omega ^\omega ,\omega }$, by a slight extension of the argument given in [33, Section 4].

The following is a restricted version:

$\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,<\omega }}$:

$$\begin{aligned}\forall \alpha [\forall \gamma \exists n[\gamma (n)\le \alpha ^{\upharpoonright 0}(n)\rightarrow \bigl (n, \gamma (n)\bigr ) \in E_{\alpha ^{\upharpoonright 1}}]\rightarrow \exists n\forall i \le \alpha ^{\upharpoonright 0}(n)[ (n,i )\in E_{\alpha ^{\upharpoonright 1}}]].\end{aligned}$$

We introduce a strong negation of this restricted version:

$\lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,<\omega }})$:

$$\begin{aligned}\exists \alpha [\forall \gamma \exists n[\gamma (n)\le \alpha ^{\upharpoonright 0}(n)\rightarrow \bigl (n,\gamma (n)\bigr ) \in E_{\alpha ^{\upharpoonright 1}}]\;\wedge \; \lnot \exists n\forall i \le \alpha ^{\upharpoonright 0}(n)[(n, i) \in E_{\alpha ^{\upharpoonright 1}}]].\end{aligned}$$

Note that $\textsf{BIM}$ proves that $\forall \alpha [\lnot \exists n\forall i\le \alpha ^{\upharpoonright 0}(n)[(n, i) \in E_\alpha ]\leftrightarrow$

$\forall n\forall t\in \omega ^{\alpha ^{\upharpoonright 0}(n)+1}\exists i\le \alpha ^{\upharpoonright 0}(n)[\alpha \bigl (t(i)\bigr ) \ne (n,i)+1]]$.

Lemma 5.5

$\textsf{BIM}$ proves:

• (i) $\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,<\omega }} \rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}}$ and $\lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}}) \rightarrow \lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,<\omega }})$

• (ii) $\textbf{FT}\rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,<\omega }}$ and $\lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,<\omega }}) \rightarrow \lnot !\textbf{FT}$.

Proof

(i) We prove, in $\textsf{BIM}$: for each $\alpha$, there exists $\beta$ such that

$$\begin{aligned}{} & {} (1)\; \forall \gamma \in 2^\omega \exists n[\bigl (n,\gamma (n)\bigr )\in E_{\alpha }]\rightarrow \forall \gamma \exists n[\gamma (n)\le \beta ^{\upharpoonright 0}(n)\rightarrow \bigl (n, \gamma (n)\bigr ) \in E_{\beta ^{\upharpoonright 1}}]\;\text {and}\\{} & {} (2)\; \exists n \forall i \le \beta ^{\upharpoonright 0}(n)[(n,i)\in E_{\beta ^{\upharpoonright 1}}]\rightarrow \exists n \forall i<2[(n,i)\in E_{\alpha }]. \end{aligned}$$

The two promised conclusions then follow easily.

Let $\alpha$ be given. Define $\beta$ such that $\beta ^ {\upharpoonright 0}=\underline{1}$ and $\beta ^{\upharpoonright 1} =\alpha$.

(1) Assume $\forall \gamma \in 2^\omega \exists n[\bigl (n,\gamma (n)\bigr )\in E_{\alpha }]$. Let $\gamma$ be given. Define $\gamma ^*$ such that $\forall n[\gamma ^*(n) =\min \bigl (1, \gamma (n)\bigr )]$. Note: $\gamma ^*\in 2^\omega$ and find n such that $\bigl (n,\gamma ^*(n) \bigr )\in E_{\alpha }$. If $\gamma ^*(n) \ne \gamma (n)$, then $\gamma (n)>1=\beta ^{\upharpoonright 0}(n)$, and if $\gamma ^*(n) =\gamma (n)$, then $\bigl (n,\gamma (n)\bigr )\in E_{\alpha }$. Conclude that $\forall \gamma \exists n[\gamma (n)\le \beta ^{\upharpoonright 0}(n)\rightarrow \bigl (n, \gamma (n)\bigr ) \in E_{\beta ^{\upharpoonright 1}}]$.

(2) Let n be given such that $\forall i \le \beta ^{\upharpoonright 0}(n)[(n,i)\in E_{\beta ^{\upharpoonright 1}}]$. Conclude that $\forall i<2[(n,i)\in E_{\alpha }]$.

(ii)¹⁴ We prove, in $\textsf{BIM}$: for each $\alpha$, there exists $\beta$ such that

$$\begin{aligned}{} & {} (1)\; \forall \gamma \exists n[\gamma (n)\le \alpha ^{\upharpoonright 0}(n)\rightarrow \bigl (n,\gamma (n)\bigr ) \in E_{\alpha ^{\upharpoonright 1}}] \rightarrow Bar_{2^\omega }(D_\beta )\;\text {and}\\{} & {} (2)\; \exists m[ Bar_{2^\omega }D_{{\overline{\beta }} m})] \rightarrow \exists n\forall m \le \alpha ^{\upharpoonright 0}(n)[(n, m) \in E_{\alpha ^{\upharpoonright 1}}]. \end{aligned}$$

The two promised conclusions then follow easily.

Define $Cod_2:\omega \rightarrow 2^{<\omega }$ such that $Cod_2(\langle \;\rangle )=\langle \; \rangle$ and $\forall s\forall n[Cod_2(s*\langle n \rangle ) = Cod_2(s)*\underline{{\overline{0}}}n *\langle 1 \rangle ]$. Note that $\forall t\in 2^{<\omega }\exists s\exists i[ t= Cod_2(s) *\underline{{\overline{0}}}i]$. Let $\alpha$ be given. Define $\beta$ such that, for all s, i, $\beta \bigl (Cod_2(s)*\underline{{\overline{0}}}i\bigr ) \ne 0$ if and only if $\exists n< \textit{length}(s)[s(n) >\alpha ^{\upharpoonright 0}(n)\;\vee \; \bigl (n,s(n)\bigr )\in E_{\overline{\alpha ^{\upharpoonright 1}}length(s)}\;\vee \;\alpha ^{\upharpoonright 0}\bigl (\textit{length}(s)\bigr )< i]$.

(1) Assume that $\forall \gamma \exists n[\gamma (n) \le \alpha ^{\upharpoonright 0}(n) \rightarrow \bigl (n,\gamma (n)\bigr ) \in E_{\alpha ^{\upharpoonright 1}}]$. Assume that $\delta \in 2^\omega$. Define $\gamma$, by induction, such that, for each n, if $\exists i\le \alpha ^{\upharpoonright 0}(n)[Cod_2({\overline{\gamma }} n*\langle i\rangle )\sqsubset \delta ]$, then $\gamma (n)=\mu i[Cod_2({\overline{\gamma }} n*\langle i\rangle )\sqsubset \delta ]$, and, if not, then $\gamma (n)=0$. Note that $\forall n[\gamma (n)\le \alpha ^{\upharpoonright 0}(n)]$. Find n, p such that $\bigl (n,\gamma (n)\bigr ) \in E_{\overline{\alpha ^{\upharpoonright 1}}p}$. Define $q:=\max (n,p)$. Note that $\beta \bigl (Cod_2({\overline{\gamma }}(q+1))\bigr )\ne 0$ and distinguish two cases. Case (a). $c:=Cod_2({\overline{\gamma }}(q+1))\sqsubset \delta$ and $\beta (c)\ne 0$. Case (b). $\exists m\le q\forall i\le \alpha ^{\upharpoonright 0}(m)[Cod_2({\overline{\gamma }} m*\langle i \rangle )\perp \delta ]$. Find $m_0:=\mu m\le q\forall i\le \alpha ^{\upharpoonright 0}(m)[Cod_2({\overline{\gamma }} m*\langle i \rangle )\perp \delta ]$ and note that $d:=Cod_2({\overline{\gamma }} m_0)*\underline{{\overline{0}}}\bigl (\alpha ^{\upharpoonright 0}(m_0)+1\bigr )\sqsubset \delta$ and $\beta (d) \ne 0$. In both cases $\exists p[\beta ({\overline{\delta }} p) \ne 0]$. We thus see that $\forall \delta \in 2^\omega \exists p[\beta ({\overline{\delta }} p)\ne 0]$, i.e. $Bar_{2^\omega }(D_\beta )$.

(2) Let m be given such that $Bar_{2^\omega }(D_{{\overline{\beta }} m})$. Suppose that

$\forall i<m \exists j \le \alpha ^{\upharpoonright 0}(i)[(i,j)\notin E_{\overline{ \alpha ^1}m}]]$. Find s such that $\textit{length}(s) = m$ and $\forall i<m[s(i)\le \alpha ^{\upharpoonright 0}(i)\;\wedge \;\bigl (i,s(i)\bigr )\notin E_{\overline{ \alpha ^{\upharpoonright 1}}m}]$. Note that $Cod_2(s)>m$ and $\forall t\sqsubseteq Cod_2(s)[\beta (t)=0 ]$, so $\lnot \textit{Bar}_{2^\omega }(D_{{\overline{\beta }} m})$. Contradiction. Conclude that

$\exists i <m\forall j\le \alpha ^{\upharpoonright 0}(i)[(i, j) \in E_{\overline{\alpha ^{\upharpoonright 1}}m}\subseteq E_{\alpha ^{\upharpoonright 1}}]$. $\square$

Theorem 5.6

$\textsf{BIM}$ proves: $\textbf{FT} \leftrightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,<\omega }}$ and $\lnot !\textbf{FT} \leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,<\omega }})$.

Proof

These statements follow from Lemma 5.5 and Theorem 5.4. $\square$

5.1 No double negation shift

Assume $\lnot !{\textbf {FT}}$. Using $\lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega , 2}})$, find $\alpha$ such that $\forall \gamma \in 2^\omega \exists n[ \bigl (n,\gamma (n)\bigr ) \in E_{\alpha }]$ and $\lnot \exists n\forall m<2[(n,m)\in E_\alpha ]$. Then, for each n, $\lnot \forall m <2[(n,m)\in E_\alpha ]$ and $\lnot \forall m<2[\lnot \lnot \bigl ((n,m)\in E_\alpha \bigr )]$ and $\lnot \lnot \exists m<2[(n,m)\notin E_\alpha ]$. Conclude that $\forall n\lnot \lnot \exists m<2[(n,m) \notin E_{\alpha }]$. Note that $\lnot \exists \gamma \in 2^\omega \forall n[ \bigl (n,\gamma (n)\bigr ) \notin E_{\alpha }]$. Using $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, conclude that $\lnot \forall n\exists m<2[(n,m)\notin E_\alpha ]$. We thus see that if we assume both $\lnot !{\textbf {FT}}$ and $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ we can find $\varvec{\Pi }^0_1$-subsets $P=\{n\mid (n,0)\notin E_\alpha \}$ and $Q=\{n\mid (n,1)\notin E_\alpha \}$ of $\omega$ such that $\forall n[\lnot \lnot \bigl (P(n) \vee Q(n)\bigr )]$ and $\lnot \forall n[P(n) \vee Q(n)]$. S. Kuroda’s scheme of Double Negation Shift $\forall n[\lnot \lnot T(n)]\rightarrow \lnot \lnot \forall n[T(n)]$ (see [19, page 45] and [8, page 105]) thus is refuted.

In [30, vol. I, Chapter 4, Proposition 3.4, Corollary 1], the same conclusion is obtained in $\textsf{HA}$ from $\textsf{CT}_0$.

6 ${\textbf {AC}}_{\omega ,\omega ^\omega }$, some special cases

6.1. $\varvec{\Sigma }^0_1$-Second axiom of countable choice, $\varvec{\Sigma }^0_1$-${\textbf {AC}}_{\omega ,\omega ^\omega }$:

$$\begin{aligned}\forall \alpha [\forall n \exists \gamma [\gamma \in \mathcal {G}_{\alpha ^{\upharpoonright n}}] \rightarrow \exists \gamma \forall n[\gamma ^{\upharpoonright n} \in \mathcal {G}_{\alpha ^{\upharpoonright n}}]].\end{aligned}$$

Theorem 6.1

$\textsf{BIM}\vdash \varvec{\Sigma }^0_1$-$\textbf{AC}_{\omega ,\omega ^\omega }$.

Proof

Assume $\forall n \exists \gamma [ \gamma \in \mathcal {G}_{\alpha ^{\upharpoonright n}}]$. Then $\forall n \exists s[\alpha ^{\upharpoonright n}( s)\ne 0]$. Find $\delta$ such that $\forall n[\delta (n)=\mu s[\alpha ^{\upharpoonright n}( s) \ne 0]$]. Find $\gamma$ such that $\forall n[\gamma ^{\upharpoonright n} = \delta (n) *\underline{0}]$. Note that $\forall n[ \gamma ^{\upharpoonright n} \in \mathcal {G}_{\alpha ^{\upharpoonright n}}]$. $\square$

6.2. $\varvec{\Pi }^0_1$-Second axiom of countable choice, $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega ^\omega }$:

$$\begin{aligned}\forall \alpha [\forall n \exists \gamma [ \gamma \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}] \rightarrow \exists \gamma \forall n [\gamma ^{\upharpoonright n} \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]].\end{aligned}$$

Theorem 6.2

$\textsf{BIM}\vdash \varvec{\Pi }^0_1$-$\textbf{AC}_{\omega ,\omega ^\omega } \rightarrow \varvec{\Pi }^0_1$-$\textbf{AC}_{\omega ,\omega }$.

Proof

Let $\alpha$ be given such that $\forall n \exists m [(n, m) \notin E_{\alpha }]$. Define $\beta$ such that

$\forall n \forall a[\beta ^{\upharpoonright n}(a) \ne 0 \leftrightarrow \exists m\exists b \exists p\le a[ \alpha (p)=(n,m)+1\;\wedge \;a=\langle m\rangle *b]]$. Note that $\forall n \forall m[(n,m)\in E_\alpha \leftrightarrow \forall \gamma [\langle m\rangle *\gamma \in \mathcal {G}_{\beta ^{\upharpoonright n}}]]$. Conclude that $\forall n \exists \gamma [\gamma \notin \mathcal {G}_{\beta ^{\upharpoonright n}}]$. Using $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega ^\omega }$, find $\gamma$ such that $\forall n[\gamma ^{\upharpoonright n} \notin \mathcal {G}_{\beta ^{\upharpoonright n}}]$. Define $\delta$ such that $\forall n[\delta (n)=\gamma ^{\upharpoonright n}(0)]$ and note: $\forall n [\bigl ( n, \delta (n)\bigr )\notin E_\alpha ]$. $\square$

One may conclude that $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega ^\omega }$ is unprovable in $\textsf{BIM}$, see Sect. 4.8.

Not every $\varvec{\Pi }^0_1$ subset of $\omega ^\omega$ is a spread, see Lemma 2.12. For spreads, which are a special kind of $\varvec{\Pi }^0_1$ sets, countable choice is easier:

Theorem 6.3

$\textsf{BIM}\vdash \forall \alpha [\bigl (\forall n[Spr(\alpha ^{\upharpoonright n})]\;\wedge \;\forall n \exists \gamma [ \gamma \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]\bigr ) \rightarrow \exists \gamma \forall n [\gamma ^{\upharpoonright n} \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]]$.

Proof

Let $\alpha$ be given such that $\forall n[ Spr(\alpha ^{\upharpoonright n})]$ and $\forall n\exists \gamma [ \gamma \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]$. Define $\gamma$ such that, for each n, for each m, $\gamma ^{\upharpoonright n}(m)=\mu k[\alpha ^{\upharpoonright n} \bigl ( (\overline{\gamma ^{\upharpoonright n}}m)*\langle k \rangle \bigr ) = 0]$. $\square$

6.3.Axiom scheme of countable compact choice, ${\textbf {AC}}_{\omega ,2^\omega }$:

$$\begin{aligned} \forall n \exists \gamma \in 2^\omega [R(n,\gamma )]\rightarrow \exists \gamma \in 2^\omega \forall n[R(n, \gamma ^{\upharpoonright n})]. \end{aligned}$$

Here is a restricted version of ${\textbf {AC}}_{\omega ,2^\omega }$:

6.4. $\varvec{\Pi }_1^0$-Axiom of countable compact choice, $\varvec{\Pi }_1^0$-${\textbf {AC}}_{\omega ,2^\omega }$:

$$\begin{aligned}\forall \alpha [ \forall n \exists \gamma \in 2^\omega [\gamma \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}] \rightarrow \exists \gamma \in 2^\omega \forall n[\gamma ^{\upharpoonright n} \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]]\end{aligned}$$

Theorem 6.4

$\textsf{BIM}\vdash \varvec{\Pi }^0_1$-AC$_{\omega ,2^\omega }\rightarrow \varvec{\Pi }^0_1$-$\textbf{AC}_{\omega , 2}$.

Proof

The proof is almost the same as the proof of Theorem 6.2 and is left to the reader. $\square$

We may conclude: $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2^\omega }$ is unprovable in $\textsf{BIM}$, see Sect. 4.8.

The treatment of real numbers in $\textsf{BIM}$ is sketched in Sect. 13.7.

6.5. $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,[0,1]}$:

$$\begin{aligned}\forall \alpha [ \forall n \exists \delta \in [0,1] [\delta \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}] \rightarrow \exists \delta \in [0,1]^\omega \forall n[\delta ^{\upharpoonright n} \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]].\end{aligned}$$

Theorem 6.5

$\textsf{BIM}\vdash \varvec{\Pi }^0_1$-AC$_{\omega ,2^\omega }\leftrightarrow \varvec{\Pi }_1^0$-AC$_{\omega ,[0,1]}$.

Proof

First assume $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2^\omega }$.

Using Lemma 13.2, find $\sigma : 2^\omega \rightarrow [0,1]$ and $\psi :\omega ^\omega \rightarrow \omega ^\omega$ such that

• (1) $\forall \delta \in [0,1] \exists \gamma \in 2^\omega [\delta =_\mathcal {R} \sigma |\gamma ]$ and

• (2) $\forall \alpha \forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_{\psi |\alpha }\leftrightarrow \sigma |\gamma \in \mathcal {H}_\alpha ]$.

Let $\alpha$ be given such that $\forall n \exists \delta \in [0,1] [\delta \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$. Then $\forall n \exists \gamma \in 2^\omega [\sigma |\gamma \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$ and $\forall n \exists \gamma \in 2^\omega [\gamma \notin \mathcal {G}_{\psi |(\alpha ^{\upharpoonright n})}]$. Find $\gamma$ in $2^\omega$ such that $\forall n[\gamma ^{\upharpoonright n}\notin \mathcal {G}_{\psi |(\alpha ^{\upharpoonright n})}]$. Conclude that $\forall n[\sigma |\gamma ^{\upharpoonright n} \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$ and $\exists \delta \forall n[\delta ^{\upharpoonright n}\notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$. We thus see that, for all $\alpha$, if $\forall n \exists \delta \in [0,1] [\delta \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$, then $\exists \delta \forall n[\delta ^{\upharpoonright n}\notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$, i.e. $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,[0,1]}$

Now assume $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,[0,1]}$.

Using Lemma 13.3, find $\tau :2^\omega \rightarrow [0,1]$ and $\chi :\omega ^\omega \rightarrow \omega ^\omega$ such that

• (1) $\forall \gamma \in 2^\omega \forall \delta \in 2^\omega [\gamma \;\#\;\delta \rightarrow \tau |\gamma \;\#_\mathcal {R}\;\tau |\delta ]$, and

• (2) $\forall \alpha \forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_\alpha \leftrightarrow \tau |\gamma \in \mathcal {H}_{\chi |\alpha }]$. and

• (3) $\forall \alpha \forall \delta \in [0,1]^\omega \exists \gamma \in 2^\omega \forall n[\delta ^{\upharpoonright n}\;\#_\mathcal {R}\;\tau |(\gamma ^{\upharpoonright n}) \rightarrow \delta ^{\upharpoonright n} \in \mathcal {H}_{\chi |(\alpha ^{\upharpoonright n})}]$.

Let $\alpha$ be given such that $\forall n \exists \gamma \in 2^\omega [\gamma \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]$. Conclude that $\forall n\exists \gamma \in 2^\omega [\tau |\gamma \notin \mathcal {H}_{\chi |(\alpha ^{\upharpoonright n})}]$ and $\forall n\exists \delta \in [0,1][\delta \notin \mathcal {H}_{\chi |(\alpha ^{\upharpoonright n})}]$. Using $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,[0,1]}$, find $\delta$ in $[0,1]^\omega$ such that $\forall n[\delta ^{\upharpoonright n} \notin \mathcal {H}_{\chi |(\alpha ^{\upharpoonright n})}]$. Using (3), find $\gamma$ in $2^\omega$ such that $\forall n[\delta ^{\upharpoonright n}\;\#_\mathcal {R}\;\tau |(\gamma ^{\upharpoonright n}) \rightarrow \delta ^{\upharpoonright n} \in \mathcal {H}_{\chi |(\alpha ^{\upharpoonright n})}]$. Conclude that $\forall n[\delta ^{\upharpoonright n}=_\mathcal {R} \tau |(\gamma ^{\upharpoonright n})]]$, and, using (2): $\forall n[\gamma ^{\upharpoonright n} \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]$. Clearly, $\exists \gamma \in 2^\omega \forall n[\gamma ^{\upharpoonright n} \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]$. We thus see that, for all $\alpha$, if $\forall n \exists \gamma \in 2^\omega [\gamma \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]$, then $\exists \gamma \in 2^\omega \forall n[\gamma ^{\upharpoonright n} \notin \mathcal {G}_{\alpha ^{\upharpoonright n}}]$, i.e. $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2^\omega }$. $\square$

7 Contrapositions of some special cases of ${{\textbf {AC}}}_{\omega ,\omega ^\omega }$

Let us consider the following axiom scheme, the Second Axiom Scheme of Reverse Countable Choice.

7.1. $\overleftarrow{{\textbf {AC}}_{\omega ,\omega ^\omega }}$: $\forall \gamma \in \omega ^\omega \exists n[R(n,\gamma ^{\upharpoonright n})]\rightarrow \exists n \forall \gamma \in \omega ^\omega [R(n,\gamma )]$.

The axiom scheme $\overleftarrow{{\textbf {AC}}_{\omega ,\omega ^\omega }}$ implies $\varvec{\Delta }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,\omega }}$ and, therefore, ${\textbf {LPO}}$, see Theorem 5.1. Let us consider a restricted version, the Axiom Scheme of Reverse Countable Compact Choice:

7.2. $\overleftarrow{{\textbf {AC}}_{\omega ,2^\omega }}$: $\forall \gamma \in 2^\omega \exists n[R(n,\gamma ^{\upharpoonright n})]\rightarrow \exists n \forall \gamma \in 2^\omega [R(n,\gamma )]$.

In [33] it is shown that $\overleftarrow{{\textbf {AC}}_{\omega ,2^\omega }}$ is a consequence of the First Axiom of Continuous Choice ${\textbf {AC}}_{\omega ^\omega ,\omega }$ and FAN.

We now require the relation R to be $\varvec{\Sigma }^0_1$ and obtain the $\varvec{\Sigma }^0_1$-Axiom of Reverse Countable Compact Choice:

7.3. $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2^\omega }}$: $\forall \alpha [ \forall \gamma \in 2^\omega \exists n [\gamma ^{\upharpoonright n} \in \mathcal {G}_{\alpha ^{\upharpoonright n}}] \rightarrow \exists n[ 2^\omega \subseteq \mathcal {G}_{\alpha ^{\upharpoonright n}}] ].$

We also introduce a strong negation:

7.4. $\lnot !\bigl (\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2^\omega }})$: $\exists \alpha [ \forall \gamma \in 2^\omega \exists n [\gamma ^{\upharpoonright n} \in \mathcal {G}_{\alpha ^{\upharpoonright n}}]\;\wedge \; \lnot \exists n[ 2^\omega \subseteq \mathcal {G}_{\alpha ^{\upharpoonright n}}] ].$

We also introduce a ‘real’ version:

7.5. $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,[0,1]}}$: $\forall \alpha [ \forall \delta \in [0,1]^\omega \exists n [\delta ^{\upharpoonright n} \in \mathcal {H}_{\alpha ^{\upharpoonright n}}] \rightarrow \exists n [ [0,1] \subseteq \mathcal {H}_{\alpha ^{\upharpoonright n}}]].$

and a strong negation:

7.6. $\lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,[0,1]}})$: $\exists \alpha [ \forall \delta \in [0,1]^\omega \exists n [\delta ^{\upharpoonright n} \in \mathcal {H}_{\alpha ^{\upharpoonright n}}] \;\wedge \; \lnot \exists n[ [0,1] \subseteq \mathcal {H}_{\alpha ^{\upharpoonright n}}]].$

The treatment of real numbers in $\textsf{BIM}$ is sketched in Sect. 13.7.

Lemma 7.1

$\textsf{BIM}$ proves:

• (i) $\textbf{FT}\rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2^\omega }}$ and $\lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2^\omega }})\rightarrow \lnot !\textbf{FT}$.

• (ii) $\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2^\omega }}\rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}}$ and $\lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}})\rightarrow \lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2^\omega }})$

• (iii) $\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}}\rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}}$ and $\lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2}})\rightarrow \lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}})$

Proof

(i)¹⁵ We prove, in $\textsf{BIM}$: for each $\alpha$, there exists $\beta$ such that

$$\begin{aligned} \forall \gamma \in 2^\omega \exists n [\gamma ^{\upharpoonright n} \in \mathcal {G}_{\alpha ^{\upharpoonright n}}] \rightarrow Bar_{2^\omega }(D_\beta )\;\text {and}\;\exists m[Bar_{2^\omega }(D_{{\overline{\beta }} m})] \rightarrow \exists n [2^\omega \subseteq \mathcal {G}_{\alpha ^{\upharpoonright n}}]. \end{aligned}$$

The two promised statements then follow easily.

Let $\alpha$ be given. Define $\beta$ such that, for every s, $\beta (s) \ne 0\leftrightarrow \bigl (s \in 2^{<\omega }\;\wedge \; \exists n < \textit{length}(s)\exists p \le \textit{length}(s^{\upharpoonright n})[\alpha ^{\upharpoonright n}( \overline{s^{\upharpoonright n}}p) \ne 0]\bigr )$.

Assume that $\forall \gamma \in 2^\omega \exists n \exists p[\alpha ^{\upharpoonright n}(\overline{\gamma ^{\upharpoonright n}} p)\ne 0]$. Conclude that $\forall \gamma \in 2^\omega \exists n[\beta ({\overline{\gamma }} n)\ne 0]$, i.e. $Bar_{2^\omega }(D_\beta )$.

Now let m be given such that $Bar_{2^\omega }(D_{{\overline{\beta }} m})$. Conclude that $\forall s \in 2^{<\omega }[s>m\rightarrow \exists t[ t\sqsubseteq s \;\wedge \; t\in D_{{\overline{\beta }} m}]]$. We have to prove that, for some n, $2^\omega \subseteq \mathcal {G}_{\alpha ^{\upharpoonright n}}$, i.e. $\forall \gamma \in 2^\omega \exists p[{\overline{\gamma }} p \in D_{\alpha ^{\upharpoonright n}}]$. We will prove the stronger statement that, for some $n<m$, $\forall u \in 2^{<\omega }[length(u)=m\rightarrow \exists p < m[ {\overline{u}} p \in D_{\alpha ^{\upharpoonright n}}]]$. We argue by contradiction. Assume that, for each $n<m$, there exists u in $2^{<\omega }$ such that $length(u)=m$ and $\lnot \exists p<m[{\overline{u}} p \in D_{\overline{\alpha ^{\upharpoonright n}}m}]$. Let s be an element of $\textit{Bin}$ such that, for each $n<m$, $s^{\upharpoonright n}\in 2^{<\omega }$ and $length(s^{\upharpoonright n})\ge m$ and $\lnot \exists p<m[\overline{s^{\upharpoonright n}}p \in D_{\overline{\alpha ^{\upharpoonright n}}m}]$. Note that $s>m$ and $\lnot \exists t\sqsubseteq s[t\in D_{{\overline{\beta }} m}]$. Contradiction. Thus we see there must exist $n<m$ such that $\forall u\in 2^{<\omega }[length(u)=m\rightarrow \exists p<m[{\overline{u}} p \in D_{\overline{\alpha ^{\upharpoonright n}}m}]]$ and $2^\omega \subseteq \mathcal {G}_{\alpha ^{\upharpoonright n}}$.

(ii)¹⁶ We prove, in $\textsf{BIM}$: for each $\alpha$, there exists $\beta$ such that

$$\begin{aligned}{} & {} \forall \delta \in [0,1]^\omega \exists n [\delta ^{\upharpoonright n} \in \mathcal {H}_{\alpha ^{\upharpoonright n}}] \rightarrow \forall \gamma \in 2^\omega \exists n [\gamma ^{\upharpoonright n} \in \mathcal {G}_{\beta ^{\upharpoonright n}}]\;\text {and}\\{} & {} \exists n [2^\omega \subseteq \mathcal {G}_{\beta ^{\upharpoonright n}}] \rightarrow \exists n [[0,1] \subseteq \mathcal {H}_{\alpha ^{\upharpoonright n}}]. \end{aligned}$$

Using Lemma 13.2, find $\sigma :2^\omega \rightarrow [0,1]$ and $\psi :\omega ^\omega \rightarrow \omega ^\omega$ such that $\forall \delta \in [0,1]\exists \gamma \in 2^\omega [\sigma |\gamma =_\mathcal {R} \delta ]$ and $\forall \alpha \forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_{\psi |\alpha }\leftrightarrow \sigma |\gamma \in \mathcal {H}_{\alpha }]$.

Let $\alpha$ be given. Define $\beta$ such that, for every n, $\beta ^{\upharpoonright n} = \psi |(\alpha ^{\upharpoonright n})$.

Assume that $\forall \delta \in [0,1]^\omega \exists n[\delta ^{\upharpoonright n} \in \mathcal {H}_{\alpha ^{\upharpoonright n}}]$. Then $\forall \gamma \in 2^\omega \exists n[\sigma |(\gamma ^{\upharpoonright n}) \in \mathcal {H}_{\alpha ^{\upharpoonright n}}]$ and $\forall \gamma \in 2^\omega \exists n[\gamma ^{\upharpoonright n} \in \mathcal {G}_{\psi |(\alpha ^{\upharpoonright n})}]$ and $\forall \gamma \in 2^\omega \exists n [\gamma ^{\upharpoonright n} \in \mathcal {G}_{\beta ^{\upharpoonright n}}]$.

Let n be given such that such that $2^\omega \subseteq \mathcal {G}_{\beta ^{\upharpoonright n}}=\mathcal {G}_{\psi |(\alpha ^{\upharpoonright n})}$. Note that $\forall \gamma \in 2^\omega [ \sigma |\gamma \in \mathcal {H}_{\alpha ^{\upharpoonright n}}]$. Conclude that $[0,1] \subseteq \mathcal {H}_{\alpha ^{\upharpoonright n}}$.

(iii) We prove in $\textsf{BIM}$ that, for each $\alpha$, there exists $\beta$ such that

$$\begin{aligned}{} & {} \forall \gamma \in 2^\omega \exists n [\bigl (n,\gamma (n)\bigr ) \in E_{\alpha }] \rightarrow \forall \delta \in [0,1]^\omega \exists n [\delta ^{\upharpoonright n} \in \mathcal {H}_{\beta ^{\upharpoonright n}}]\;\text {and}\\{} & {} \exists n [[0,1] \subseteq \mathcal {H}_{\beta ^{\upharpoonright n}}] \rightarrow \exists n \forall i<2[(n,i)\in E_{\alpha }]. \end{aligned}$$

Let $\alpha$ be given. Define $\beta$ such that $\forall n\forall s\in \mathbb {S}[\beta ^{\upharpoonright n}(s) \ne 0\leftrightarrow \exists i<s[\bigl (\alpha (i) =(n,0)+1\;\wedge \;s''<_\mathbb {Q}1_\mathbb {Q}\bigr )\;\vee \;\bigl (\alpha (i) =(n,1)+1\;\wedge \;0_\mathbb {Q}<_\mathbb {Q}s'\bigr )]]$.

Note that $\forall n[\bigl ((n,0) \in E_{\alpha }\leftrightarrow [0,1) \subseteq \mathcal {H}_{\beta ^{\upharpoonright n}}]\bigr )\;\wedge \;\bigl ((n,1) \in E_{\alpha }\leftrightarrow (0,1] \subseteq \mathcal {H}_{\beta ^{\upharpoonright n}}\bigr )]$.

Assume that $\forall \gamma \in 2^\omega \exists n [ \bigl (n,\gamma (n)\bigr ) \in E_{\alpha }]$, and $\delta \in [0,1]^\omega$. Define $\varepsilon$ such that $\forall n[\varepsilon (n)=\mu m [0_\mathbb {Q}<_\mathbb {Q}\bigl (\delta ^{\upharpoonright n}(m)\bigr )'\;\vee \;\bigl (\delta ^{\upharpoonright n}(m)\bigr )''<_\mathbb {Q}1_\mathbb {Q}]$. Define $\gamma$ in $2^\omega$ such that $\forall n[\gamma (n) = 0\leftrightarrow \bigl (\delta ^{\upharpoonright n}(\varepsilon (n))\bigr )''<_\mathbb {Q}1_\mathbb {Q}]$. Note that $\forall n[\gamma (n) = 1\rightarrow 0_\mathcal {R}<_\mathcal {R} \delta ^{\upharpoonright n} ]$. Find n such that $\bigl (n,\gamma (n)\bigr ) \in E_{\alpha }$ and conclude that either $\gamma (n) = 0$ and $\delta ^{\upharpoonright n} <_\mathcal {R} 1_\mathcal {R}$ and $[0,1) \subseteq \mathcal {H}_{\beta ^{\upharpoonright n}}$, so $\delta ^{\upharpoonright n} \in \mathcal {H}_{\beta ^{\upharpoonright n}}$, or $\gamma (n) =1$ and $0_\mathcal {R} < _\mathcal {R}\delta ^{\upharpoonright n}$ and $(0,1]\subseteq \mathcal {H}_{\beta ^{\upharpoonright n}}$, so, again, $\delta ^{\upharpoonright n} \in \mathcal {H}_{\beta ^{\upharpoonright n}}$. Conclude that $\forall \delta \in [0,1]^\omega \exists n[\delta ^{\upharpoonright n} \in \mathcal {H}_{\beta ^{\upharpoonright n}}]$.

Let n be given such that $[0,1] \subseteq \mathcal {H}_{\beta ^{\upharpoonright n}}$. Conclude that $\forall i<2[(n,i)\in E_{\alpha }]$. $\square$

Theorem 7.2

• (i) $\textsf{BIM}\vdash \textbf{FT} \leftrightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2^\omega }}\leftrightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}}$.

• (ii) $\textsf{BIM}\vdash \lnot !\textbf{FT} \leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,2^\omega }})\leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}})$.

Proof

These statements follow from Lemmas 7.1 and  5.3.$\square$

8 On the contraposition of twofold compact choice

We introduce a limited version of $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2^\omega }}$:

8.1. $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$: $\forall \alpha [ \forall \gamma \in 2^\omega \exists i<2 [\gamma ^{\upharpoonright i} \in \mathcal {G}_{\alpha ^i}] \rightarrow \exists i<2 [2^\omega \subseteq \mathcal {G}_{\alpha ^{\upharpoonright i}}] ].$

This statement should be called the $\varvec{\Sigma }^0_1$-Axiom of Reverse Twofold Compact Choice. It is a contraposition of a special case of the following scheme:

$$\begin{aligned}\forall i<2 \exists \gamma \in 2^\omega [R(i,\gamma )] \rightarrow \exists \gamma \in 2^\omega \forall i<2[R(i,\gamma ^{\upharpoonright i})].\end{aligned}$$

and the latter scheme is provable in $\textsf{BIM}$.

For each $\alpha$, we define the following statement, called ${\textbf {LLPO}}^\alpha$:

$$\begin{aligned}{} & {} \forall \varepsilon [\forall p [2p=\mu m[\varepsilon (m)\ne 0] \rightarrow \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p})]\;\vee \;\\{} & {} \forall p [2p+1=\mu m[\varepsilon (m)\ne 0] \rightarrow \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p})]]. \end{aligned}$$

Lemma 8.1

• (i) $\textsf{BIM}\vdash \textbf{LLPO}\leftrightarrow \forall \alpha [\textbf{LLPO}^\alpha ]$.

• (ii) $\textsf{BIM}\;+\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{2,2^\omega }} \vdash \forall \alpha [Bar_{2^\omega }(D_\alpha )\rightarrow \textbf{LLPO}^\alpha ]$.

• (iii) $\textsf{BIM}\;+$ $\varvec{\Pi }^0_1$-$\textbf{AC}_{\omega ,2}\vdash \forall \alpha [Bar_{2^\omega }(D_\alpha )\rightarrow \textbf{LLPO}^\alpha ]\rightarrow \textbf{FT}$.

• (iv) $\textsf{BIM}\vdash \textbf{FT}\rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{2,2^\omega }}$.

Proof

(i) Assume ${\textbf {LLPO}}$ and let $\alpha , \varepsilon$ be given. Either $\forall p[2p\ne \mu n[\varepsilon (n)\ne 0]]$ and, therefore, $\forall \varepsilon [\forall p [2p=\mu m[\varepsilon (m)\ne 0] \rightarrow \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p})]$, or $\forall p[2p+1\ne \mu n[\varepsilon (n)\ne 0]]$ and $\forall \varepsilon [\forall p [2p+1=\mu m[\varepsilon (m)\ne 0] \rightarrow \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p})]$. We thus see: ${\textbf {LLPO}}^\alpha$.

For the converse, note that ${\textbf {LLPO}}\leftrightarrow {\textbf {LLPO}}^{\underline{0}}$.

(ii) Let $\alpha$ be given such that $Bar_{2^\omega }(D_\alpha )$. Using $\varvec{\Sigma }_1^0$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$, we will prove ${\textbf {LLPO}}^\alpha$. Let $\varepsilon$ be given. Define $\eta$ such that, for each p,

• (1) if $\underline{{\overline{0}}}(2p+2)\sqsubset \varepsilon$, then $\eta ^{\upharpoonright 0}(p) = \eta ^{\upharpoonright 1}(p) = \alpha (p)$, and,

• (2) if $2p =\mu m [\varepsilon (m) \ne 0 ]$, then $\forall m \ge p[\eta ^{\upharpoonright 0}(m) =0\;\wedge \;\eta ^{\upharpoonright 1}(m) = \alpha (m)]$, and

• (3) if $2p+1=\mu m[\varepsilon (m) \ne 0 ]$, then $\forall m \ge p[\eta ^{\upharpoonright 1}(m) =0\;\wedge \;\eta ^{\upharpoonright 0}(m) = \alpha (m)]$.

Note that, if $\eta ^{\upharpoonright 0} \;\#\; \alpha$, then $\eta ^{\upharpoonright 1} = \alpha$.

Let $\gamma$ in $2^\omega$ be given. Find n such that $\alpha (\overline{\gamma ^{\upharpoonright 0}}n) \ne 0$. Either $\eta ^{\upharpoonright 0}(\overline{\gamma ^{\upharpoonright 0}}n) = \alpha (\overline{\gamma ^{\upharpoonright 0}}n)\ne 0$, or $\eta ^{\upharpoonright 0} \;\#\; \alpha$ and $\eta ^{\upharpoonright 1} = \alpha$ and $\exists m[\eta ^{\upharpoonright 1}(\overline{\gamma ^{\upharpoonright 1}}m) = \alpha (\overline{\gamma ^{\upharpoonright 1}}m) \ne 0]$. We thus see that $\forall \gamma \in 2^\omega [\gamma ^{\upharpoonright 0} \in \mathcal {G}_{\eta ^{\upharpoonright 0}} \;\vee \;\gamma ^{\upharpoonright 1} \in \mathcal {G}_{\eta ^{\upharpoonright 1}}]$. Use $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$ and find $i<2$ such that $2^\omega \subseteq \mathcal {G}_{\eta ^{\upharpoonright i}}$, i.e. $Bar_{2^\omega }(D_{\eta ^{\upharpoonright i}})$. Let p be given such that $2p+i= \mu m [\varepsilon (m) \ne 0]$. Note that $\forall m\ge p[\eta ^{\upharpoonright i}(m) = 0]$. Conclude that $\textit{Bar}_{2^\omega }(D_{\overline{\eta ^{\upharpoonright i}} p})$ and that $\textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p})$. We thus see that $\forall p[2p+i= \mu m [\varepsilon (m) \ne 0] \rightarrow \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p})]$. Conclude that $\exists i<2\forall p[2p+i= \mu m [\varepsilon (m) \ne 0] \rightarrow \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p})]$, i.e. ${\textbf {LLPO}}^\alpha$.

(iii) Assume $\forall \alpha [Bar_{2^\omega }(D_\alpha )\rightarrow {\textbf {LLPO}}^\alpha ]$. Using $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, we will prove ${\textbf {FT}}$.

Let $\alpha$ be given such that $Bar_{2^\omega }(D_\alpha )$ and, therefore, ${\textbf {LLPO}}^\alpha$. Let s in $2^{<\omega }$ be given. Define $\varepsilon$ such that, $\forall i<2\forall n[\varepsilon (2n+i) \ne 0 \leftrightarrow Bar_{2^\omega \cap s*\langle i \rangle }(D_{{\overline{\alpha }} n})]$. Using ${\textbf {LLPO}}^\alpha$, find $i<2$ such that $\forall p[2p+i= \mu m [\varepsilon (m) \ne 0] \rightarrow \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p})]$. Assume we find n such that $Bar_{2^\omega \cap s*\langle i\rangle }(D_{{\overline{\alpha }} n})$. Then $\varepsilon (2n+i) \ne 0$. Find $p:= \mu j[\varepsilon (j) \ne 0]$. Find $q\le n$ such that $p=2q$ or $p=2q+1$. Either $p=2q+i$ and $\textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} q})$, or $p=2q+1-i$ and $Bar_{2^\omega \cap s*\langle 1-i\rangle } (D_{{\overline{\alpha }} q})$. In both cases, $Bar_{2^\omega \cap s*\langle 1-i\rangle }(D_{{\overline{\alpha }} n})$. We thus see that $\forall n[Bar_{2^\omega \cap s*\langle i \rangle } (D_{{\overline{\alpha }} n}) \rightarrow Bar_{2^\omega \cap s*\langle 1-i \rangle } (D_{{\overline{\alpha }} n}) ]$. Conclude that $\forall s \in 2^{<\omega }\exists i<2\forall n[ Bar_{2^\omega \cap s*\langle i \rangle }( D_{{\overline{\alpha }} n})\rightarrow Bar_{2^\omega \cap s*\langle 1- i \rangle } (D_{{\overline{\alpha }} n})]$. Now use $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ and find $\gamma$ in $2^\omega$ such that $\forall s \in 2^{<\omega } \forall n[ Bar_{2^\omega \cap s*\langle \gamma (s) \rangle }( D_{{\overline{\alpha }} n})\rightarrow Bar_{2^\omega \cap s*\langle 1-\gamma (s) \rangle } (D_{{\overline{\alpha }} n})]$. Observe that, for each s in $2^{<\omega }$, for all n, if $Bar_{2^\omega \cap s*\langle \gamma (s) \rangle }( D_{{\overline{\alpha }} n})$, then also

$Bar_{2^\omega \cap s*\langle 1- \gamma (s) \rangle }( D_{{\overline{\alpha }} n})$, and, therefore, $Bar_{2^\omega \cap s}( D_{{\overline{\alpha }} n})$. Define $\delta$ in $2^\omega$ such that, for each n, $\delta (n) = \gamma (\overline{ \delta } n)$. Find p such that $\alpha (\overline{ \delta } p) \ne 0$ and define $n:={\overline{\delta }} p +1$. Note that $Bar_{2^\omega \cap {\overline{\delta }} p}(D_{{\overline{\alpha }} n})$. One now proves, by backwards induction, that, for each $j \le p$, $Bar_{2^\omega \cap {\overline{\delta }} j}(D_{{\overline{\alpha }} n})$. The induction step goes as follows. Assume $j+1 \le n$ and $Bar_{2^\omega \cap \overline{\delta } (j+1)}(D_{{\overline{\alpha }} n})$. As $\overline{ \delta }(j+1) = \overline{ \delta } j *\langle \gamma (\overline{\delta } j)\rangle$, one conclude that $Bar_{2^\omega \cap \overline{\delta } (j)}(D_{{\overline{\alpha }} n})$. After n steps one sees $Bar_{2^\omega }(D_{{\overline{\alpha }} n})$. Conclude that $\forall \alpha [Bar_{2^\omega }(D_\alpha )\rightarrow \exists n[Bar_{2^\omega }(D_{{\overline{\alpha }} n})]]$, i.e. ${\textbf {FT}}$.

(iv) Assume ${\textbf {FT}}$. Use Theorem 7.2 and conclude $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2^\omega }}$ and its corollary $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$. $\square$

Theorem 8.2

$\textsf{BIM}+\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}\vdash \varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}\leftrightarrow {\textbf {FT}}$.

Proof

Use Lemma 8.1. $\square$

8.1.1. Mark Bickford called my attention to the fact that $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2, 2^\omega }}$ occurs in [22, §2] and is called there the separation principle $\text {SP}$.

After having proved ${\textbf {FT}}\rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$, see our Lemma 8.1(iv), the author of [22] gives a proof of ${\textbf {FT}}\rightarrow {\textbf {WKL!}}$ using $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }!$ and quotes the result ${\textbf {WKL}}!\rightarrow {\textbf {FT}}$, see our Theorem 2.7. Our proof of Theorem 2.8 shows that no choice is needed for a proof of ${\textbf {FT}}\rightarrow {\textbf {WKL!}}$.

We introduce a ‘real’ version of $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$:

8.2. $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,[0,1]}}$:

$$\begin{aligned}\forall \alpha [ \forall \delta \in [0,1]^2 \exists i<2 [\delta ^{\upharpoonright i} \in \mathcal {H}_{\alpha ^{\upharpoonright i}}] \rightarrow \exists i<2 [ [0,1] \subseteq \mathcal {H}_{\alpha ^{\upharpoonright i}}] ].\end{aligned}$$

Theorem 8.3

$\textsf{BIM}\vdash \varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}\leftrightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,[0,1]}}$.

Proof

Using Lemma 13.2, find $\sigma : 2^\omega \rightarrow [0,1]$ and $\psi : \omega ^\omega \rightarrow \omega ^\omega$ such that

$\forall \delta \in [0,1] \exists \gamma [\delta =_\mathcal {R} \sigma |\gamma ]$ and $\forall \alpha \forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_{\psi |\alpha }\leftrightarrow \sigma |\gamma \in \mathcal {H}_\alpha ]$.

First assume $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$. Let $\alpha$ be given such that $\forall \delta \in [0,1]^2\exists i<2[\delta ^{\upharpoonright i} \in \mathcal {H}_{\alpha ^{\upharpoonright i}}]$. Define $\beta$ such that, for both $i<2$, $\beta ^{\upharpoonright i} = \psi |(\alpha ^{\upharpoonright i})$. Then $\forall \gamma \in 2^\omega \exists i < 2[\sigma |\gamma ^{\upharpoonright i} \in \mathcal {H}_{\alpha ^{\upharpoonright i}}]$. Conclude that $\forall \gamma \in 2^\omega \exists i <2[\gamma ^{\upharpoonright i} \in \mathcal {G}_{\beta ^{\upharpoonright i}}]$. Find $i <2$ such that $2^\omega \subseteq \mathcal {G}_{\beta ^{\upharpoonright i}}$. Conclude $[0,1]\subseteq \mathcal {H}_{\alpha ^{\upharpoonright i}}$. We thus see $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,[0,1]}}$.

Now assume $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,[0,1]}}$. Using Lemma 13.3, find $\tau :2^\omega \rightarrow [0,1]$ and $\chi :\omega ^\omega \rightarrow \omega ^\omega$ such that

• (1) $\forall \gamma \in 2^\omega \forall \delta \in 2^\omega [\gamma \;\#\;\delta \rightarrow \tau |\gamma \;\#_\mathcal {R}\;\tau |\delta ]$, and

• (2) $\forall \alpha \forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_\alpha \leftrightarrow \tau |\gamma \in \mathcal {H}_{\chi |\alpha }]$ and

• (3) $\forall \delta \in [0,1]\exists \gamma \in 2^\omega [\delta \;\#_\mathcal {R}\;\tau |\gamma \rightarrow \forall \alpha [\delta \in \mathcal {H}_{\chi |\alpha }]]$.

Let $\alpha$ be given such that $\forall \gamma \in 2^\omega \exists i < 2[\gamma ^{\upharpoonright i} \in \mathcal {G}_{\alpha ^{\upharpoonright i}} ]$. Let $\delta$ in $[0,1]^2$ be given. Find $\gamma$ in $2^\omega$ such that $\forall i<2[\delta ^{\upharpoonright i}\;\#_\mathcal {R}\;\tau |(\gamma ^{\upharpoonright i}) \rightarrow \delta ^{\upharpoonright i}\in \mathcal {H}_{\chi |(\alpha ^{\upharpoonright i})}]$. Find $i<2$ such that $\gamma ^{\upharpoonright i}\in \mathcal {G}_{\alpha ^{\upharpoonright i}}$. Conclude that $\tau |(\gamma ^{\upharpoonright i})\in \mathcal {H}_{\chi |\alpha ^{\upharpoonright i}}$. Find s, n such that $(\chi |\alpha ^{\upharpoonright i})(s)\ne 0$ and $\bigl (\tau |(\gamma ^{\upharpoonright i})\bigr )(n)\sqsubset _\mathbb {S} s$. Using Lemma 13.1, find p such that either $\delta ^{\upharpoonright i}(p)\sqsubset _\mathbb {S} s$, and $\delta ^{\upharpoonright i}\in \mathcal {H}_{\chi |(\alpha ^i)}$, or $\delta ^{\upharpoonright i}(p)\;\#_\mathbb {S}\; \bigl (\tau |(\gamma ^{\upharpoonright i})\bigr )(n)$, and, again, $\delta ^{\upharpoonright i}\in \mathcal {H}_{\chi |(\alpha ^i)}$. We thus see that $\forall \delta \in [0,1]^2\exists i<2[\delta ^{\upharpoonright i} \in \mathcal {H}_{\chi |(\alpha ^{\upharpoonright i})}]$. Applying $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,[0,1]}}$, find $i<2$ such that $[0,1]\subseteq \mathcal {H}_{\chi |(\alpha ^{\upharpoonright i})}$. Conclude that $\forall \gamma \in 2^\omega [\tau |\gamma \in \mathcal {H}_{\chi |(\alpha ^{\upharpoonright i})}]$ and $2^\omega \subseteq \mathcal {G}_{\alpha ^{\upharpoonright i}}$. We thus see $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$. $\square$

Corollary 8.4

$\textsf{BIM} + \varvec{\Pi }^0_1$-$\textbf{AC}_{\omega ,2}\vdash \textbf{FT} \leftrightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{2,[0,1]}}$.

Proof

Use Theorems 8.2 and 8.3. $\square$

8.1 Some logical consequences

In this Subsection, we want to formulate the result of Theorem 8.2 in model-theoretic terms and draw an even sharper conclusion.

Tarski’s truth definition makes sense intuitionistically as well as classically. For every structure $\mathfrak {A}=(A, \ldots )$, for every formula $\varphi =\varphi (\mathsf {x_0, x_1, \ldots , x_{n-1}})$ in the elementary language of the structure $\mathfrak {A}$, for all $a_0, a_1, \ldots , a_{n-1}$ in A,

$$\begin{aligned}\mathfrak {A}\models \varphi [a_0, a_1, \dots , a_{n-1}]\end{aligned}$$

if and only if

the formula $\varphi$ is true in the structure $\mathfrak {A}$, provided we interpret the individual variables $\mathsf {x_0, x_1, \ldots , x_{n-1}}$ by $a_0, a_1, \ldots , a_{n-1}$, respectively.

For every structure $\mathfrak {A}=(A, \ldots )$, for every sentence $\varphi$ in the elementary language of the structure $\mathfrak {A}$,

$$\begin{aligned}\mathfrak {A}\models \varphi \end{aligned}$$

if the sentence $\varphi$ is true in the structure $\mathfrak {A}$.

For every $\delta$, we define a proposition $Pr_\delta$, as follows. $Pr_\delta := \exists n[\delta (n) \ne 0]$.

Theorem 8.5

The following statements are equivalent in $\textsf{BIM}$:

• (i) $\forall \alpha [(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x [P(x) \vee A] \rightarrow (\forall x[P(x)] \vee A)}$.

• (ii) $\forall \alpha [(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, \mathcal {G}_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x[P(x) \vee Q(x)] \rightarrow (\forall x[P(x)] \vee \exists x[Q(x)])}$.

Proof

(i) $\Rightarrow$ (ii). Let $\alpha$ be given such that $(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, \mathcal {G}_{\alpha ^{\upharpoonright 1}}) \models \mathsf {\forall x [P(x) \vee Q(x)]}$. Note that $(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, \mathcal {G}_{\alpha ^{\upharpoonright 1}}) \models \mathsf {\forall x [P(x) \vee \exists y[Q(y)]]}$. Define $\beta$ such that $\beta ^{\upharpoonright 0} = \alpha ^{\upharpoonright 0}$ and $\forall n[\beta ^{\upharpoonright 1}(n)\ne 0\leftrightarrow \bigl (\alpha ^{\upharpoonright 1}(n)\ne 0 \;\wedge \;n\in 2^{<\omega }\bigr )]$. Note that $(2^\omega , \mathcal {G}_{\beta ^{\upharpoonright 0}}, Pr_{\beta ^{\upharpoonright 1}}) \models \mathsf {\forall x [P(x) \vee A]}$. Using (i), conclude that $(2^\omega , \mathcal {G}_{\beta ^{\upharpoonright 0}}, Pr_{\beta ^{\upharpoonright 1}}) \models \mathsf {\forall x [P(x)] \vee A}$ and that $(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, \mathcal {G}_{\alpha ^{\upharpoonright 1}}) \models \mathsf {\forall x [P(x)] \vee \exists x[Q(x)]}$.

(ii) $\Rightarrow$ (i). Let $\alpha$ be given such that $(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}}) \models \mathsf {\forall x[P(x) \vee A]}$. Define $\beta$ such that $\beta ^{\upharpoonright 0} = \alpha ^{\upharpoonright 0}$ and $\forall n[\beta ^{\upharpoonright 1}(n)\ne 0\leftrightarrow \bigl (\exists i<n[\alpha ^{\upharpoonright 1}(i) \ne 0] \;\wedge \;n \in 2^{<\omega }\bigr )]$. Note that, for each $\gamma$ in $2^\omega$, $\gamma \in \mathcal {G}_{\beta ^{\upharpoonright 1}}$ if and only if $Pr_{\alpha ^{\upharpoonright 1}}$. Conclude that $(2^\omega , \mathcal {G}_{\beta ^{\upharpoonright 0}}, \mathcal {G}_{\beta ^{\upharpoonright 1}}) \models \mathsf {\forall x[P(x) \vee Q(x)]}$ and, using (ii), that $(2^\omega , \mathcal {G}_{\beta ^{\upharpoonright 0}}, \mathcal {G}_{\beta ^{\upharpoonright 1}})\models \mathsf {\forall x[P(x)] \vee \exists x[Q(x)]}$ and that $(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}}) \models \mathsf {\forall x[P(x)] \vee A}$. $\square$

For each $\alpha$, we define the following statement:

$$\begin{aligned}{} {\textbf {LPO}}^\alpha :\;\forall \varepsilon [\forall p[\overline{\underline{0}}(2p+2)\perp \varepsilon \rightarrow \textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p} )]\;\vee \;\exists n[\varepsilon (n) \ne 0]].\end{aligned}$$

Lemma 8.6

• (i) $\textsf{BIM}\vdash \textbf{LPO}\rightarrow \forall \alpha [\textbf{LPO}^\alpha ]$.

• (ii) $\textsf{BIM}\vdash \forall \alpha [\textbf{LPO}^\alpha \rightarrow \textbf{LLPO}^\alpha ]$.

Proof

The proof is left to the reader. $\square$

Theorem 8.7

The following statements are equivalent in $\textsf{BIM}\;+\;\varvec{\Pi }^0_1$-$\textbf{AC}_{\omega ,2}$:

• (i) ${\textbf {FT}}$.

• (ii) $\forall \alpha [(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, \mathcal {G}_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x \forall y[P(x) \vee Q(y)] \rightarrow (\forall x[P(x)] \vee \forall y[Q(y)])}]$.

• (iii) $\forall \alpha [(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x [P(x) \vee A] \rightarrow (\forall x[P(x)] \vee A)}]$.

• (iv) $\forall \alpha [Bar_{2^\omega }(D_\alpha )\rightarrow {\textbf {LPO}}^\alpha ]$.

Proof

(i) $\Rightarrow$ (ii). Note that, in $\textsf{BIM} + \varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega , 2}$, ${\textbf {FT}}$ implies $\varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{2,2^\omega }}$:

$\forall \alpha [ \forall \gamma \in 2^\omega \exists i<2 [\gamma ^{\upharpoonright i} \in \mathcal {G}_{\alpha ^{\upharpoonright i}}] \rightarrow \exists i<2 [2^\omega \subseteq \mathcal {G}_{\alpha ^{\upharpoonright i}}] ]$, see Theorem 8.2.

(ii) $\Rightarrow$ (iii). Let $\alpha$ be given such that $(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x [P(x) \vee A]}$, i.e. $\forall \gamma \in 2^\omega \exists n \exists i<2[\alpha ^{\upharpoonright 0}({\overline{\gamma }} n)\ne 0\;\vee \; \alpha ^{\upharpoonright 1}(n)\ne 0]$. Define $\beta$ such that $\beta ^{\upharpoonright 0}=\alpha ^{\upharpoonright 0}$ and $\forall s \in 2^{<\omega }[\beta ^{\upharpoonright 1}(s)= \alpha ^{\upharpoonright 1}(n)]$. Note $(2^\omega , \mathcal {G}_{\beta ^{\upharpoonright 0}}, \mathcal {G}_{\beta ^{\upharpoonright 1}})\models \mathsf {\forall x \forall y[P(x) \vee Q(y)]}$ and conclude, using (i), $(2^\omega , \mathcal {G}_{\beta ^{\upharpoonright 0}}, \mathcal {G}_{\beta ^{\upharpoonright 1}})\models \mathsf {\forall x[P(x)] \vee \forall y[Q(y)]}$ and $(2^\omega , \mathcal {G}_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}})\models \mathsf { \forall x[P(x)] \vee A}$.

(iii) $\Rightarrow$ (iv). Assume (iii).

Let $\alpha$ be given such that $\textit{Bar}_{2^\omega }(D_\alpha )$. We have to prove ${\textbf {LPO}}^\alpha$. Let $\varepsilon$ be given. Define $\eta$ in $2^\omega$ such that $\eta ^{\upharpoonright 1} = \varepsilon$, and, for each p, if $\underline{{\overline{0}}}(2p+2)\sqsubset \varepsilon$, then $\eta ^{\upharpoonright 0}(p) = \alpha (p)$, and, if $\underline{{\overline{0}}}(2p+2)\perp \varepsilon$, then $\eta ^{\upharpoonright 0}(p) =0$. Note that, if $\eta ^{\upharpoonright 0} \;\#\; \alpha$, then $\exists n[\varepsilon (n) =\eta ^{\upharpoonright 1} (n) \ne 0]$. Let $\gamma$ in $2^\omega$ be given. Find n such that $\alpha ({\overline{\gamma }} n) \ne 0$. Either $\eta ^0(\overline{\gamma }n) = \alpha (\overline{\gamma }n)\ne 0$ or $\eta ^{\upharpoonright 0} \;\#\; \alpha$ and $\exists m[\eta {\upharpoonright 1}(m) \ne 0]$. We thus see that $\forall \gamma \in 2^\omega [\exists n[\eta ^{\upharpoonright 0}(\overline{\gamma }n) \ne 0]\;\vee \;\exists m[\eta ^{\upharpoonright 1}(m) \ne 0]]$. Use (iii) and conclude that $\forall \gamma \in 2^\omega \exists n[\eta ^{\upharpoonright 0}({\overline{\gamma }} n) = 1]\;\vee \;\exists m[\eta ^{\upharpoonright 1}(m) \ne 0]$. First, assume $\forall \gamma \in 2^\omega \exists n[\eta ^{\upharpoonright 0}({\overline{\gamma }} n) \ne 0]$ i.e. $\textit{Bar}_{2^\omega }(D_{\eta ^{\upharpoonright 0}})$. Let p be given such that $\underline{{\overline{0}}}(2p+2)\perp \varepsilon$. Note that $\forall m\ge p[\eta ^{\upharpoonright 0}(m) = 0]$. Conclude that $\textit{Bar}_{2^\omega }(D_{\overline{\eta ^{\upharpoonright 0}} p})$ and $\textit{Bar}_{2^\omega }(D_{{\overline{\alpha }} p})$. We thus see that $\forall p[ \underline{{\overline{0}}}(2p+2)\perp \varepsilon \rightarrow \textit{Bar}_2^\omega (D_{{\overline{\alpha }} p})]$. Secondly, assume $\exists n[\eta ^{\upharpoonright 1}(n)\ne 0]$. Conclude that $\exists n [\varepsilon (n) \ne 0]$. We thus see that $\forall \varepsilon [\overline{\underline{0}}(2p+2)\perp \varepsilon \rightarrow Bar_2^\omega (D_{{\overline{\alpha }} p})]\;\vee \;\exists n[\varepsilon (n)\ne 0]]$, i.e. ${\textbf {LPO}}^\alpha$.

(iv) $\Rightarrow$ (i). Use Lemma 8.6(ii) and Theorem 8.2(iii). $\square$

The sentences mentioned in Theorems 8.5 and 8.7 are true in every structure $(\{0, 1, \ldots , n\}, P, Q, A)$ where n is a natural number, P, Q are arbitrary subsets of $\{0,1, \ldots , n\}$ and A is an arbitrary proposition, that is, these sentences hold in every finite structure. They sometimes fail to be true in countable structures, as appears from the next two theorems.

Theorem 8.8

The following statements are equivalent in $\textsf{BIM}$.

• (i) $\textbf{LLPO}$.

• (ii) $\forall \alpha [(\omega ,D_{\alpha ^{\upharpoonright 0}}, D_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x \forall y[P(x) \vee Q(y)]}\rightarrow \mathsf {(\forall x[P(x)] \vee \forall y [Q(y)])}$.

Proof

(i) $\Rightarrow$ (ii). Let $\alpha$ be given. Define $\beta$ such that $\forall p\forall i<2[\beta (2p+i) = 0\leftrightarrow \alpha ^{\upharpoonright i}(p)\ne 0]$. Assume that $(\omega , D_{\alpha ^{\upharpoonright 0}}, D_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x \forall y[P(x) \;\vee \; Q(y)]}$. Conclude that $\forall p \forall q[\beta (2p) = 0 \;\vee \;\beta (2q+1) = 0]$. Apply ${\textbf {LLPO}}$ and distinguish two cases.

Case (1). $\forall p[2p+1\ne \mu m[\beta (m)\ne 0]]$. Assume we find p such that $\beta (2p+1)\ne 0$. Determine q such that $q\le p$ and $\beta (2q)\ne 0$. Contradiction. Conclude that $\forall p[\beta (2p+1)=0]$ and $D_{\alpha ^{\upharpoonright 1}} = \omega$.

Case (2). $\forall p[2p\ne \mu m[\beta (m)\ne 0]]$. Then, for similar reasons, $D_{\alpha ^{\upharpoonright 0}} = \omega$.

In both cases $(\omega ,D_{\alpha ^{\upharpoonright 0}}, D_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x[P(x)] \vee \forall y [Q(y)]}$.

(ii) $\Rightarrow$ (i). Assume (ii). Let $\alpha$ be given. Define $\beta$ such that, for each p, for both $i<2$, $\beta ^{\upharpoonright i}(p) = 0$ if and only if $2p+i = \mu n[\alpha (n)\ne 0]$. Note that $(\omega , D_{\beta ^{\upharpoonright 0}}, D_{\beta ^{\upharpoonright 1}}) \models \mathsf {\forall x \forall y[P(x) \;\vee \; Q(y)]}$. Conclude that $(\omega , D_{\beta ^{\upharpoonright 0}}, D_{\beta ^{\upharpoonright 1}}) \models \mathsf {\forall x [P(x)] \;\vee \; \forall y[Q(y)]}$. Conclude that either $(\omega , D_{\beta ^{\upharpoonright 0}}, D_{\beta ^{\upharpoonright 1}}) \models \mathsf {\forall x [P(x)] }$ and $\forall p[2p\ne \mu n[\alpha (n)\ne 0]]$, or

$(\omega , D_{\beta ^{\upharpoonright 0}}, D_{\beta ^{\upharpoonright 1}}) \models \mathsf {\forall y [Q(y)] }$ and $\forall p[2p+1\ne \mu n[\alpha (n)\ne 0]]$. Conclude ${\textbf {LLPO}}$. $\square$

Theorem 8.9

The following statements are equivalent in $\textsf{BIM}$.

• (i) ${\textbf {LPO}}$.

• (ii) $\forall \alpha [(\omega ,D_{\alpha ^{\upharpoonright 0}}, D_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x [P(x) \vee Q(x)]}\rightarrow \mathsf {(\forall x[P(x)] \vee \exists x [Q(x)])}$.

• (iii) $\forall \alpha [(\omega ,D_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x [P(x) \vee A]}\rightarrow \mathsf {(\forall x[P(x)] \vee A)}$.

Proof

(i) $\Rightarrow$ (ii). Let $\alpha$ be given such that $(\omega ,D_{\alpha ^{\upharpoonright 0}}, D_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x [P(x) \vee Q(x)]}$, i.e. $\forall n[\alpha ^{\upharpoonright 0}(n)\ne 0\;\vee \;\alpha ^{\upharpoonright 1}(n)\ne 0]$. Using ${\textbf {LPO}}$, distinguish two cases. Either $\forall n[\alpha ^{\upharpoonright 0}(n) \ne 0]$ and $(\omega ,D_{\alpha ^{\upharpoonright 0}}, D_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x[P(x)]}$, or $\exists n[\alpha ^{\upharpoonright 0}(n) =0]$ and $\exists n[\alpha ^{\upharpoonright 1}(n) \ne 0]$ and $(\omega ,D_{\alpha ^{\upharpoonright 0}}, D_{\alpha ^{\upharpoonright 1}})\models \mathsf {\exists x[Q(x)]}$.

(ii) $\Rightarrow$ (i). Let $\alpha$ be given. Define $\beta$ such that $\forall n[\beta ^{\upharpoonright 0}(n) = 0\leftrightarrow \alpha (n) \ne 0]$ and $\beta ^{\upharpoonright 1}=\alpha$. Note that $\forall n[\beta ^{\upharpoonright 0}(n)\ne 0\;\vee \;\beta ^1(n)\ne 0]$, i.e. $(\omega , D_{\beta ^{\upharpoonright 0}}, D_{\beta ^{\upharpoonright 1}})\models \mathsf {\forall x[P(x)\;\vee \;Q(x)]}$. Use (ii) and distinguish two cases. Either $(\omega , D_{\beta ^{\upharpoonright 0}}, D_{\beta ^{\upharpoonright 1}})\models \mathsf {\forall x[P(x)]}$ and $\forall n[\alpha (n) = 0]$, or $(\omega , D_{\beta ^{\upharpoonright 0}}, D_{\beta ^{\upharpoonright 1}})\models \mathsf {\exists x[Q(x)]}$ and $\exists n[\alpha (n) \ne 0]$. Conclude $\forall \alpha [\forall n[\alpha (n)=0]\;\vee \;\exists n[\alpha (n)\ne 0]]$, i.e. ${\textbf {LPO}}$.

(i) $\Rightarrow$ (iii). Let $\alpha$ be given such that $(\omega ,D_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x [P(x) \vee A]}$, i.e. $\forall n[\alpha ^{\upharpoonright 0}(n)\ne 0\;\vee \;\exists m[\alpha ^{\upharpoonright 1}(m)\ne 0]]$. Using ${\textbf {LPO}}$, distinguish two cases. Either $\forall n[\alpha ^{\upharpoonright 0}(n) \ne 0]$ and $(\omega ,D_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x[P(x)]}$, or $\exists n[\alpha ^{\upharpoonright 0}(n) =0]$ and $\exists m[\alpha ^{\upharpoonright 1}(m) \ne 0]$ and $(\omega ,D_{\alpha ^{\upharpoonright 0}}, Pr_{\alpha ^{\upharpoonright 1}})\models \textsf{A}$.

(iii) $\Rightarrow$ (i). Let $\alpha$ be given. Define $\beta$ such that $\forall n[\beta (n)=0 \leftrightarrow \alpha (n)\ne 0]$. Note that $\forall n[\beta (n)\ne 0\;\vee \; \alpha (n)\ne 0]$, i.e. $(\omega , D_\beta , Pr_\alpha )\models \mathsf {\forall x[P(x)\;\vee \;A]}$. Use (iii) and distinguish two cases. Either $(\omega , D_\beta , Pr_\alpha )\models \mathsf {\forall x[P(x)]}$ and $\forall n[\beta (n)\ne 0]$ and $\forall n[\alpha (n) = 0]$, or $(\omega , D_\beta , Pr_\alpha )\models \textsf{A}$ and $\exists n[\alpha (n) \ne 0]$. Conclude $\forall \alpha [\forall n[\alpha (n)=0]\;\vee \;\exists n[\alpha (n)\ne 0]]$, i.e. ${\textbf {LPO}}$. $\square$

8.2 A note

According to Theorem 8.7(iii), $\textsf{BIM}+\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ proves that ${\textbf {FT}}$ is equivalent to:

For all $\alpha$, if $\forall \gamma \in 2^\omega \exists n[ \alpha ^{\upharpoonright 0}({\overline{\gamma }} n) \ne 0 \vee \alpha ^{\upharpoonright 1}(n) \ne 0]$,

then either $\forall \gamma \in 2^\omega \exists n[\alpha ^{\upharpoonright 0}({\overline{\gamma }} n) \ne 0]$ or $\exists n[\alpha ^{\upharpoonright 1}(n) \ne 0]$.

One might ask if $\textsf{BIM}+\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$ proves that $\lnot !{\textbf {FT}}$ is equivalent to the following statement:

There exists $\alpha$ such that $\forall \gamma \in 2^\omega \exists n[ \alpha ^{\upharpoonright 0}({\overline{\gamma }} n) \ne 0 \vee \alpha ^{\upharpoonright 1}(n) \ne 0]$ and $\lnot \forall \gamma \in 2^\omega \exists n[\alpha ^{\upharpoonright 0}({\overline{\gamma }} n) \ne 0]$ and $\alpha ^{\upharpoonright 1}=\underline{0},$

i.e.

$(*)$: There exists $\alpha$ such that

$\forall \gamma \in 2^\omega \exists n[ \alpha ({\overline{\gamma }} n) \ne 0]$ and $\lnot \forall \gamma \in 2^\omega \exists n[\alpha ({\overline{\gamma }} n) \ne 0]$.

The formula $(*)$ is an outright contradiction!

9 The determinacy of finite and infinite games

We consider games of perfect information for players I, II. First finite games, then games with finitely many moves where the players may choose out of infinitely many alternatives, and then games of infinite length. In the last Subsection we prove that, in $\textsf{BIM}$, ${\textbf {FT}}$ is an equivalent of the Intuitionistic Determinacy Theorem. This theorem says that every subset of $(\omega \times 2)^\omega$ is weakly determinate.

9.1 Finite games

9.1.1 Finite choice and a contraposition of finite choice

Lemma 9.1

$\textsf{BIM}$ proves the following scheme:

$$\begin{aligned} \forall m[\forall n<m[A(n)\vee B]\rightarrow (\forall n<m[A(n)]\;\vee \; B)] \end{aligned}$$

Proof

The proof is straightforward, by induction. $\square$

Lemma 9.2

$\textsf{BIM}$ proves the following schemes:

• (i) $\forall k[\forall n< k\exists m[R(n,m)] \rightarrow \exists s \forall n< k[R\bigl (n,s(n)\bigr )]]$.

• (ii) $\forall k\forall l[\forall s:k \rightarrow l\exists n< k[R\bigl (n,s(n)\bigr )] \rightarrow \exists n< k \forall m< l[R(n,m)]]$.

Proof

(i) The proof is by induction on k and left to the reader.

(ii) The proof is by induction on k and uses Lemma 9.1. Note that there is nothing to prove if $k=0$. Now assume the statement holds for a certain k.

Assume $\forall s:(k+1)\rightarrow l\exists n < k+1[ R\bigl (n,s(n)\bigr )]$. Note that

$$\begin{aligned} \forall j<l\forall s:k \rightarrow l [\exists n < k[R\bigl (n, s(n)\bigr )]\;\vee \;R(k,j) ], \end{aligned}$$

and, therefore, by Lemma 9.1,

$$\begin{aligned} \forall j<l[R(k,j)\;\vee \;\forall s:k \rightarrow l \exists n < k[R\bigl (n, s(n)\bigr )] ]. \end{aligned}$$

Using the induction hypothesis, we conclude that

$$\begin{aligned} \forall j<l[R(k,j)]\;\vee \;\exists n<k\forall m<l[ R(n,m)], \end{aligned}$$

i.e. $\exists n< k +1\forall m<l [R(n,m)]$. $\square$

9.1.2 Finite games

We want to study finite and infinite games for players I and II of perfect information. We first consider finite games. In such games, there are finitely many moves, and for each move there are only finitely many alternatives.

Assume $X\subseteq \omega$. Let n, l be given such that $l>0$.

Players I and II play the I-game for X in $\textit{Seq}(n,l)$ and the II-game for X in $\textit{Seq}(n,l)$ in the same way, as follows. First, player I chooses $i_0<l$, then player II chooses $i_1<l$, and they continue until a finite sequence $\langle i_0, i_1, \ldots , i_{n-1}\rangle$ of length n has been formed, a so-called final position. Player I wins the play $\langle i_0, i_1, \ldots , i_{n-1}\rangle$ in the I-game for X if and only if $\langle i_0, i_1, \ldots , i_{n-1} \rangle \in X$. Player II wins the play $\langle i_0, i_1, \ldots , i_{n-1}\rangle$ in the II-game for X if and only if $\langle i_0, i_1, \ldots , i_{n-1} \rangle \in X$.

We define $\varvec{\varphi }$ such that, for all n, for all $l>0$,

$$\begin{aligned} \varvec{\varphi }(n,l):=\mu a\forall i< n\forall c \in Seq(i, l)[c<a]. \end{aligned}$$

Every number coding a position in Seq(n, l) that is not a final position is smaller than $\varvec{\varphi }(n,l)$.

We define $\varvec{\psi }$ such that, for all n, for all $l>0$,

$$\begin{aligned} \varvec{\psi }(n,l):=\mu a\forall s\in Seq\bigl (\varvec{\varphi }(n,l),l\bigr )[s<a]. \end{aligned}$$

When studying games in Seq(n, l) it suffices to consider strategies s, t for player I, II, respectively, such that $s,t<\varvec{\psi }(n,l)$.

The reader should consult Sect. 13.10 for some of the notations we are going to use, like ‘$\in _I$’ and ‘$\in _{II}$’.

We define: $X\subseteq \omega$ is I-determinate in $\textit{Seq}(n,l)$, $\textit{Det}^I_{\textit{Seq}(n,l)}(X)$,

if and only if $\forall t < \varvec{\psi }(n,l)\exists c [c \in _{II}t \;\wedge \; c \in X]\rightarrow \exists s\forall c \in \textit{Seq}(n,l)[c \in _I s \rightarrow c \in X],$ and: $X\subseteq \omega$ is II-determinate in $\textit{Seq}(n,l)$, $\textit{Det}^{II}_{\textit{Seq}(n,l)}(X)$,

if and only if $\forall s < \varvec{\psi }(n,l)\exists c [c \in _{I}s \;\wedge \; c \in X]\rightarrow \exists t\forall c \in \textit{Seq}(n,l)[c \in _{II} t \rightarrow c \in X].$

Theorem 9.3

(Determinacy of finite games) Let $X\subseteq \omega$ be given.

For every $l>0$, for every n, $\textit{Det}^I_{\textit{Seq}(n,l)}(X)$ and $\textit{Det}^{II}_{\textit{Seq}(n,l)}(X)$.

Proof

Let $X\subseteq \omega$ and $l>0$ be given.

For every u, we define: $X_{u}:=\{c \in \omega \mid u*c \in X\}$.

We intend to prove a statement seemingly stronger than the statement of the theorem:

$(*)$: for every n, for every u, $\textit{Det}^I_{\textit{Seq}(n,l)}(X_u)$ and $\textit{Det}^{II}_{\textit{Seq}(n,l)}(X_u)$.

The proof is by induction on n. If $n=0$, then, for every u, the statements: ‘$X_u$ is I-determinate in $\textit{Seq}(0,l)$’ and ‘$X_u$ is II-determinate in $\textit{Seq}(0,l)$’ both assert: ‘if $\langle \; \rangle \in X_u$, then $\langle \; \rangle \in X_u$’, and thus are obviously true.

Now assume the statement $(*)$ has been established for a certain n. We prove that $(*)$ is also true for $n+1$.

Let u be given.

First, assume that $\forall t <\varvec{\psi }(n+1,l)\exists c[c \in _{II} t\;\wedge \; c \in X_u]$, or, equivalently,¹⁷$\forall t<\varvec{\psi }(n+1,l) \exists k <l\exists c[c \in _I t^{\upharpoonright k} \;\wedge \; \langle k \rangle *c \in X_u ]$. Note that $\forall k<l\forall c[\langle k\rangle *c \in X_u\leftrightarrow c \in X_{u*\langle k\rangle }]$. Conclude that $\forall t: l \rightarrow \psi (n,l) \exists k<l\exists c[c \in _I t(k) \;\wedge \; c \in X_{u*\langle k \rangle }]$. Use Lemma 9.2(ii) and find $k<l$ such that $\forall s <\varvec{\psi }(n,l)\exists c[c \in _I s \; \wedge \; c \in X_{u*\langle k \rangle }]$. Use the induction hypothesis and find $t <\varvec{\psi }(n,l)$ such that $\forall c \in \textit{Seq}(n,l)[ c \in _{II}t \rightarrow c \in X_{u*\langle k \rangle }]$. Define $s<\varvec{\psi }(n+1,l)$ such that $s(\langle \; \rangle )= k$ and $s^{\upharpoonright k} = t$. Note that $\forall c\in \textit{Seq}(n+1,l)[ c \in _I s\rightarrow c \in X_u]$. We thus see that $X_u$ is I-determinate.

Next, assume that $\forall s <\varvec{\psi }(n+1,l)\exists c[c \in _{I} s\;\wedge \; c \in X_u]$, or, equivalently, $\forall s <\varvec{\psi }(n+1,l) \exists c[c \in _{II} s^{\upharpoonright s(\langle \; \rangle )} \;\wedge \; \langle s(\langle \; \rangle ) \rangle *c \in X_u ]$. Note that $\forall k<l \forall t<\varvec{\psi }(n,l)\exists s<\psi (n+1,l)[s(\langle \; \rangle ) = k\;\wedge \;s^{\upharpoonright k} = t]$. Conclude that $\forall k<l\forall t <\varvec{\psi }(n,l) \exists c[ c \in _{II} t \; \wedge \; \langle k \rangle *c \in X_u]$. Note that $\forall k<l\forall c[\langle k\rangle *c \in X_u\leftrightarrow c \in X_{u*\langle k\rangle }]$. Conclude that $\forall k<l\forall t <\varvec{\psi }(n,l) \exists c[ c \in _{II} t \; \wedge \; c \in X_{u*\langle k \rangle }]$. Use the induction hypothesis and conclude that $\forall k<l \exists s <\varvec{\psi }(n,l)\forall c\in \textit{Seq}(n,l)[ c \in _I s\rightarrow s \in X_{u*\langle k \rangle }]$. Use Lemma 9.2(i) and find $t<\psi (n+1,l)$ such that $\forall k<l \forall c\in \textit{Seq}(n,l)[ c \in _I t^{\upharpoonright k}\rightarrow c \in X_{u*\langle k\rangle }]$. Note that $\forall c \in \textit{Seq}(n+1,l)[ c \in _{II} t\rightarrow c \in X_u]$. We thus see that $X_u$ is II-determinate. $\square$

9.1.3 Comparison with the classical theorem

Note that, in classical mathematics, the I-determinacy of finite games is stated as follows:

For every $X\subseteq \omega$, for every $l>0$, for every n,

either $\exists t <\varvec{\psi }(n,l)\forall c \in \textit{Seq}(n,l)[c \in _{II}t \rightarrow c \notin X]$,

or $\exists s <\varvec{\psi }(n,l)\forall c \in \textit{Seq}(n,l) [c \in _I s \rightarrow c \in X]$,

i.e. either player II has a strategy ensuring that the result of the game will not be in X, or player I has a strategy ensuring that it does.

Taken constructively, this statement fails to be true already in the case $n=0$, because it then implies: for every subset X of $\{\langle \;\rangle \}$, either $\langle \;\rangle \notin X$ or $\langle \;\rangle \in X$, and therefore, for any proposition P, $\lnot P \vee P$, the principle of the excluded third.

9.2 Infinitely many alternatives

9.2.1 Infinitely many alternatives for player II

Let $X\subseteq 2 \times \omega$ be given. Players I and II play the I-game for X in $2\times \omega$ in the following way. First, player I chooses $i<2$, then player II chooses n and the play is finished. Player I wins the play $\langle i, n \rangle$ if and only if $\langle i,n\rangle \in X$.

We define: X is I-determinate in $2 \times \omega$, $\textit{Det}^I_{2 \times \omega }(X)$, if and only if

$$\begin{aligned} \forall t\exists c\in 2 \times \omega [c \in _{II}t \;\wedge \; c \in X]\rightarrow \exists s < 2\forall n[\langle s, n \rangle \in X]. \end{aligned}$$

Theorem 9.4

$\textsf{BIM}\vdash \forall \alpha [\textit{Det}^I_{2 \times \omega }(D_\alpha )]\leftrightarrow \textbf{LLPO}$.

Proof

This Theorem is a reformulation of Theorem 8.8. In order to see this, make two observations.

(i) For each $\alpha$, there exists $\beta$ such that $Det^I_{2\times \omega }(D_\alpha )\leftrightarrow (\omega ,D_{\beta ^{\upharpoonright 0}}, D_{\beta ^{\upharpoonright 1}})\models \mathsf {\forall x \forall y[P(x) \vee Q(y)]}\rightarrow \mathsf {(\forall x[P(x)] \vee \forall y [Q(y)])}$. Given any $\alpha$, define $\beta$ such that, for each n, $\beta ^{\upharpoonright 0}(n)=\alpha (\langle 0, n\rangle )$ and $\beta ^{\upharpoonright 1}(n)=\alpha (\langle 1, n\rangle )$.

(ii) For each $\alpha$, there exists $\beta$ such that $\bigl ((\omega ,D_{\alpha ^{\upharpoonright 0}}, D_{\alpha ^{\upharpoonright 1}})\models \mathsf {\forall x \forall y[P(x) \vee Q(y)]}\rightarrow \mathsf {(\forall x[P(x)] \vee \forall y [Q(y)])}\bigr )\leftrightarrow Det^I_{2\times \omega }(D_\beta )$. Given any $\alpha$, define $\beta$ such that, for each n, $\beta (\langle 0, n\rangle )=\alpha ^{\upharpoonright 0}(n)$ and $\beta (\langle 1, n\rangle )=\alpha ^{\upharpoonright 1}(n)$. $\square$

9.2.2 Infinitely many alternatives for player I

Let $X\subseteq \omega \times 2$ be given. Players I and II play the I-game for X in $\omega \times 2$ in the following way. First, player I chooses a natural number n, then player II chooses a number i from $\{0,1\}$. Player I wins the play $\langle n, i\rangle$ if and only if $\langle n,i\rangle \in X$.

Note that a strategy for player I in such a two-move-game coincides with his first move and thus is a natural number. A strategy for player II, on the other hand, is an infinite sequence $\tau$ in $2^\omega$ that expresses player II’s intention to play $\tau (\langle n \rangle )$ once player I has brought them to the position $\langle n \rangle$.

We define: $X\subseteq \omega \times 2$ is I-determinate in $\omega \times 2$, $\textit{Det}^I_{\omega \times 2}(X)$, if and only if:

$$\begin{aligned} \forall \tau \in 2^\omega \exists c \in \omega \times 2[c \in _{II} \tau \;\wedge \; c \in X] \rightarrow \exists s\forall i<2[ \langle s, i\rangle \in X]. \end{aligned}$$

Theorem 9.5

$\textsf{BIM}\vdash \forall \alpha [Det^I_{\omega \times 2}(D_\alpha )]$.

Proof

Let $\alpha$ be given. Assume that $\forall \tau \in 2^\omega \exists n[\langle n, \tau (\langle n \rangle )\rangle \in D_\alpha ]$. Find $\tau$ in $2^\omega$ such that $\forall n[\tau (\langle n \rangle ) = 1\leftrightarrow \langle n,0\rangle \in D_\alpha ]$. Find n such that $\langle n, \tau (\langle n\rangle ) \rangle \in D_\alpha$. Note that $\tau (\langle n\rangle ) = 1$ and $\forall i<2[\langle n,i \rangle \in D_\alpha ]$. $\square$

We define: $X\subseteq \omega \times \omega$ is II-determinate in $\omega \times \omega$, $\textit{Det}^{II}_{\omega \times \omega }(X)$, if and only if:

$$\begin{aligned} \forall m \exists n [\langle m,n\rangle \in X]\rightarrow \exists \tau \forall m[\langle m, \tau (\langle m\rangle )\rangle \in X]. \end{aligned}$$

Theorem 9.6

$\textsf{BIM}\vdash \forall \alpha [Det^{II}_{\omega \times \omega }(E_\alpha )]$.

Proof

Let $\alpha$ be given such that $\forall m\exists n [\langle m, n\rangle \in E_\alpha ]$, that is

$\forall m\exists n \exists p[\alpha (p)=\langle m, n\rangle +1 ]$. Find $\gamma$ such that $\forall m[\alpha \bigl (\gamma '(m)\bigr )=\langle m, \gamma ''(m)\rangle +1 ]$. Define $\tau$ such that $\forall m[\tau (\langle m \rangle )=\gamma ''(m)]$. $\square$

We define: $X\subseteq 2\times \omega$ is II-determinate in $2 \times \omega$, $\textit{Det}^{II}_{2 \times \omega }(X)$, if and only if:

$$\begin{aligned} \forall m<2 \exists n [\langle m,n\rangle \in X]\rightarrow \exists t\forall m<2[\langle m, t(\langle m\rangle )\rangle \in X]. \end{aligned}$$

Note that $\textsf{BIM}$ proves the scheme ${Det}^{II}_{2 \times \omega }(X)$.

9.2.3 Longer games

We also consider games in which players I, II make more than one move. Which of those games are determinate from the viewpoint of Player I? Because of Theorem 9.4, we restrict ourselves to games in which player I has, for each one of his moves, countably many alternatives, whereas player II always has to choose one of two possibilities.

For every n, for every $X\subseteq (\omega \times 2)^n$, we define:

X is I-determinate in $(\omega \times 2)^n$, $\textit{Det}_{(\omega \times 2)^n}(X)$, if and only if

$$\begin{aligned} \forall \tau \in 2^\omega \exists c[c \in _{II}\tau \;\wedge \; c \in X]\rightarrow \exists s\forall c \in (\omega \times 2)^n[c \in _I s \rightarrow c \in X]. \end{aligned}$$

This definition extends in the obvious way to subsets X of $(\omega \times 2)^n \times \omega$.

9.3 Infinitely many moves

We also want to consider games of infinite length. We imagine players I, II to build together an infinite sequence $\gamma$ in $\omega ^\omega$, as follows. First, player I chooses $\gamma (0)$, then player II chooses $\gamma (1)$, then player I chooses $\gamma (2)$, and so on.

We define a number of notions of determinacy.

$\mathcal {X}\subseteq (\omega \times 2)^\omega$ is I-determinate in $(\omega \times 2)^\omega$, $\textit{Det}^{I}_{(\omega \times 2)^\omega }(\mathcal {X})$, if and only if

$$\begin{aligned} \forall \tau \in 2^\omega \exists \gamma [\gamma \in _{II}\tau \;\wedge \; \gamma \in \mathcal {X}]\rightarrow \exists \sigma \forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \gamma \in \mathcal {X}]. \end{aligned}$$

$\mathcal {X}\subseteq (\omega \times 2)^\omega$ is finitely I-determinate in $(\omega \times 2)^\omega$, $^*\textit{Det}^{I}_{(\omega \times 2)^\omega }(\mathcal {X})$ if and only if

$$\begin{aligned} \forall \tau \in 2^\omega \exists \gamma [\gamma \in _{II}\tau \;\wedge \; \gamma \in \mathcal {X}]\rightarrow \exists s\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _{I}s \rightarrow \gamma \in \mathcal {X}]. \end{aligned}$$

$\mathcal {X}\subseteq 2^\omega$ is I-determinate in $2^\omega$, $\textit{Det}^{I}_{2^\omega }(\mathcal {X})$, if and only if

$$\begin{aligned} \forall \tau \in 2^\omega \exists \gamma [\gamma \in _{II}\tau \;\wedge \; \gamma \in \mathcal {X}]\rightarrow \exists \sigma \in 2^\omega \forall \gamma \in 2^\omega [\gamma \in _{I}\sigma \rightarrow \gamma \in \mathcal {X}]. \end{aligned}$$

$\mathcal {X} \subseteq 2^\omega$ is finitely I-determinate in $2^\omega$, $^*\textit{Det}^{I}_{2^\omega }(\mathcal {X})$, if and only if

$$\begin{aligned} \forall \tau \in 2^\omega \exists \gamma [\gamma \in _{II}\tau \;\wedge \; \gamma \in \mathcal {X}]\rightarrow \exists s\in 2^{<\omega }\forall \gamma \in 2^\omega [\gamma \in _{I}s \rightarrow \gamma \in \mathcal {X}]. \end{aligned}$$

$\mathcal {X}\subseteq 2^\omega$ is II-determinate in $2^\omega$, $\textit{Det}^{II}_{2^\omega }(\mathcal {X})$, if and only if

$$\begin{aligned} \forall \sigma \in 2^\omega \exists \gamma [\gamma \in _{I}\sigma \;\wedge \; \gamma \in \mathcal {X}]\rightarrow \exists \tau \in 2^\omega \forall \gamma \in 2^\omega [\gamma \in _{II}\tau \rightarrow \gamma \in \mathcal {X}]. \end{aligned}$$

$\mathcal {X} \subseteq 2^\omega$ is finitely II-determinate in $2^\omega$, $^*\textit{Det}^{II}_{2^\omega }(\mathcal {X})$, if and only if

$$\begin{aligned} \forall \sigma \in 2^\omega \exists \gamma [\gamma \in _{I}\sigma \;\wedge \; \gamma \in \mathcal {X}]\rightarrow \exists t\in 2^{<\omega }\forall \gamma \in 2^\omega [\gamma \in _{II}t \rightarrow \gamma \in \mathcal {X}]. \end{aligned}$$

We are going to study the following statements:

$\varvec{\Sigma }^0_1$-$\textit{Det}_{\omega \times 2}^I$: $\forall \alpha [Det^I(E_\alpha )]$.

$\varvec{\Delta }^0_1$-$\textit{Det}^I_{\omega \times 2 \times \omega }$: $\forall \alpha [ \textit{Det}^I_{\omega \times 2\times \omega }(D_\alpha )].$

$\varvec{\Delta }^0_1$-$\textit{Det}^I_{(\omega \times 2)^m}$: $\forall \alpha [ \textit{Det}^I_{(\omega \times 2)^m}(D_\alpha )].$

$\varvec{\Sigma }^0_1$-$\textit{Det}^I_{(\omega \times 2)^\omega }$: $\forall \alpha [ \textit{Det}^I_{(\omega \times 2)^\omega }(\mathcal {G}_\alpha )].$

$\varvec{\Sigma }^0_1$-$^{*}{} \textit{Det}^I_{(\omega \times 2)^\omega }$: $\forall \alpha [^{*}{} \textit{Det}^I_{(\omega \times 2)^\omega }(\mathcal {G}_\alpha )].$

$\varvec{\Sigma }^0_1$-$\textit{Det}^I_{2^\omega }$: $\forall \alpha [ \textit{Det}^I_{2^\omega }(\mathcal {G}_\alpha )].$

$\varvec{\Sigma }^0_1$-$^{*}{} \textit{Det}^I_{2^\omega }$: $\forall \alpha [ ^{*}{} \textit{Det}^I_{2^\omega }(\mathcal {G}_\alpha )].$

$\varvec{\Sigma }^0_1$-$\textit{Det}^I_{2^\omega }$: $\forall \alpha [ \textit{Det}^{II}_{2^\omega }(\mathcal {G}_\alpha )].$

$\varvec{\Sigma }^0_1$-${^{*}}{} \textit{Det}^{II}_{2^\omega }$: $\forall \alpha [{^{*}}{} \textit{Det}^{II}_{2^\omega }(\mathcal {G}_\alpha )].$

Each of the above formulas X has the form: $\forall \alpha [P(\alpha )\rightarrow Q(\alpha )]$. For each of these nine formulas X, we define the statement $\lnot ! X$, the strong negation of X, as follows:

$$\begin{aligned}\lnot ! X:=\lnot !\bigl (\forall \alpha [P(\alpha )\rightarrow Q(\alpha )]\bigr ):= \exists \alpha [P(\alpha ) \;\wedge \; \lnot Q(\alpha )].\end{aligned}$$

Note that these strong negations contain the negation symbol $\lnot$, a possibility we mentioned in Sect. 1.4.

9.4 Simulating a game in $(\omega \times 2)^{\omega }$ by a game in $2^\omega$

From the point of view of player I, a game in $(\omega \times 2)^\omega$ may be simulated by a game in Cantor space $2^\omega$. Where player I would play n in $(\omega \times 2)^\omega$, he will play n times 0 and one time 1 in $2^\omega$. So he plays the finite sequence $\underline{{\overline{0}}}n*\langle 1 \rangle$. Every time he plays 0, he makes what we call a postponing move. Player II has to react, in $2^\omega$, to these postponing moves of player I, but these reactions do not matter. As soon as player I plays 1 and completes $\overline{\underline{0}} n*\langle 1 \rangle$, player II gives, in the play in $2^\omega$, the answer he would give to player I’s move n in $(\omega \times 2)^\omega$. The reader should keep this in mind when reading the following definitions.

Define $Bin:= 2^{<\omega }= \bigcup _n \{{\overline{\gamma }} n\mid \gamma \in 2^\omega \}$, the set of (code numbers) of finite binary sequences.

Define $Halfbin:= (\omega \times 2)^{<\omega }\cup \bigl ((\omega \times 2)^{<\omega }\times \omega \bigr )=\bigcup _n\{{\overline{\gamma }} n\mid \gamma \in (\omega \times 2)^\omega \}$.

Define $\varvec{\pi _{bin}}$ in $Bin^\omega$ such that

• 1. $\varvec{\pi _{bin}}(\langle \;\rangle )=\langle \;\rangle$, and,

• 2. for each c, if length(c) is even, then, for each n, $\varvec{\pi _{bin}}(c*\langle n \rangle )=\varvec{\pi _{bin}}(c) *\underline{{\overline{0}}}2n*\langle 1\rangle$, and, for both $i<2$, $\varvec{\pi _{bin}}(c*\langle n, i\rangle )=\varvec{\pi _{bin}}(c*\langle n \rangle )*\langle i \rangle$.

The function $\varvec{\pi _{bin}}$ associates to every position in Halfbin a position in Bin.

Note that, for each c, $length(\varvec{\pi _{bin}}(c))\ge length(c)$.

Define $\varvec{\rho _{bin}}$ in $\omega ^\omega$ such that

• 1. $\varvec{\rho _{bin}}(\langle \;\rangle )=\langle \;\rangle$, and,

• 2. for each d in Bin, if length(d) is even, then $\varvec{\rho _{bin}}(d*\langle 0\rangle )=\varvec{\rho _{bin}}(d*\langle 0,0\rangle )=\varvec{\rho _{bin}}(d*\langle 0,1\rangle )=\varvec{\rho _{bin}}(d)$, and $\varvec{\rho _{bin}}(d*\langle 1\rangle )=\varvec{\rho _{bin}}(d)*\langle n \rangle$, and, for both $i<2$, $\varvec{\rho _{bin}}(d*\langle 1, i\rangle )=\varvec{\rho _{bin}}(d)*\langle n, i\rangle$, where n satisfies: either $2n=length(d)$ and $\forall i<n[d(2i)=0]$, or for some $k>0$, $length(d)=2k + 2n$ and $d(2k-2)=1$ and $\forall i<n[d(2k +2i)=0]$.

The function $\varvec{\rho _{bin}}$ associates to every position in Bin a position in Halfbin.

Note that, for every c in Halfbin, $\varvec{\rho _{bin}}\circ \varvec{\pi _{bin}}(c)=c$.

Note that, for each c in Halfbin, length(c) is even if and only if $length\bigl (\varvec{\pi _{bin}}(c)\bigr )$ is even.

Lemma 9.7

The following is provable in $\textsf{BIM}$.

For each $\alpha$, there exists $\beta$ such that

$$\begin{aligned}{} & {} \forall \tau \in 2^\omega \exists \gamma \in (\omega \times 2)^\omega [\gamma \in _{II}\tau \;\wedge \;\gamma \in \mathcal {G}_\alpha ]\\{} & {} \rightarrow \forall \tau \in 2^\omega \exists \delta \in 2^\omega [\delta \in _{II}\tau \;\wedge \;\delta \in \mathcal {G}_\beta ]\;\text {and}\\{} & {} \exists \sigma \forall \delta \in 2^\omega [\delta \in _I \sigma \rightarrow \delta \in \mathcal {G}_\beta ]\rightarrow \exists \sigma \forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \gamma \in \mathcal {G}_\alpha ]\;\text {and}\\{} & {} \exists s\forall \delta \in 2^\omega [\delta \in _I s\rightarrow \delta \in \mathcal {G}_\beta ]\rightarrow \exists s\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I s\rightarrow \gamma \in \mathcal {G}_\alpha ]. \end{aligned}$$

Proof

Let $\alpha$ be given. Define $\beta :=\alpha \circ \varvec{\rho _{bin}}$.

Assume $\forall \tau \in 2^\omega \exists \gamma \in (\omega \times 2)^\omega [\gamma \in _{II}\tau \;\wedge \;\gamma \in \mathcal {G}_\alpha ]$. Let $\tau$ be given as a strategy for player II in $2^\omega$. We have to prove that $\exists \delta \in 2^\omega [\delta \in _{II}\tau \;\wedge \; \delta \in \mathcal {G}_\beta ]$. We first define $\tau ^\dag$ as a strategy for player II in $(\omega \times 2)^\omega$. We define $\tau ^\dag$ on all positions in Halfbin of odd length, by induction on the length of the position. It suffices to define $\tau ^\dag$ on positions c such that $c \in _{II}\tau ^\dag$. We take care that, for each c in Halfbin, if $c\in _{II}\tau ^\dag$, then there exists d in Bin such that $\varvec{\rho _{bin}}(d)=c$ and $d\in _{II}\tau$.

We first define $\tau ^\dag$ on positions of length 1. Let n be given. We have to define $\tau ^\dag (\langle n \rangle )$. Find d in Bin such that $length(d)=2n+1$ and $d\in _{II} \tau$ and $d(2n)=1$ and $\forall i<n[d(2i)=0]$. Define $\tau ^\dag (\langle n \rangle ):= \tau (d)$. Note that $\varvec{\rho _{bin}}(d)=\langle n\rangle$ and $\varvec{\rho _{bin}}\bigl (d*\langle \tau (d)\rangle \bigr )=\langle n, \tau ^\dag (\langle n \rangle )\rangle$.

Now suppose $k>0$. Let c in $(\omega \times 2)^k$ be given such that $c\in _{II}\tau ^\dag$. Let n be given. We have to define $\tau ^\dag (c*\langle n \rangle )$. First find d in Bin such that $d\in _{II}\tau$ and $\varvec{\rho _{bin}}(d)=c$. Find $l:=length(d)$. Find e in Bin such that $d\sqsubset e$ and $length(e)=l + 2n+1$ and $e\in _{II} \tau$ and $e(l+2n)=1$ and $\forall i<n[e(l+2i)=0]$. Note that $\varvec{\rho _{bin}}(e)=c*\langle n\rangle$. Define $\tau ^\dag (c*\langle n \rangle ):= \tau (e)$. Note that $\varvec{\rho _{bin}}(e)= c*\langle n \rangle$ and $\varvec{\rho _{bin}}\bigl (e*\langle \tau (e)\rangle \bigr )=c*\langle \tau ^\dag (c)\rangle$.

This completes the definition of $\tau ^\dag$.

Using our assumption, find $\gamma$ in $(\omega \times 2)^\omega$ such that $\gamma \in _{II}\tau ^\dag \;\wedge \; \gamma \in \mathcal {G}_\alpha$. Note that $\forall n \exists d \in Bin[d\in _{II}\tau \;\wedge \; \varvec{\rho _{bin}}(d)={\overline{\gamma }} n]$. Note that $\forall d \in Bin\forall e \in Bin[(d\in _{II}\tau \;\wedge \;e\in _{II}\tau )\rightarrow \bigl (d\sqsubset e\leftrightarrow \varvec{\rho _{bin}}(d)\sqsubset \varvec{\rho _{bin}}(e)\bigr )]$. Using this fact, construct $\delta$ in $2^\omega$ such that $\delta \in _{II}\tau$ and $\forall n\exists m[{\overline{\gamma }} n= \varvec{\rho _{bin}}({\overline{\delta }} m)]$. Find n such that ${\overline{\gamma }} n\in D_\alpha$. Find m such that ${\overline{\gamma }} n=\varvec{\rho _{bin}}({\overline{\delta }} m)$. Note that ${\overline{\gamma }} n = \varvec{\rho _{bin}}({\overline{\delta }} m)\in D_\alpha$ and ${\overline{\delta }} m \in D_\beta$ and $\delta \in \mathcal {G}_\beta$. We thus see that $\forall \tau \in 2^\omega \exists \delta \in 2^\omega [\delta \in _{II}\tau \;\wedge \;\delta \in \mathcal {G}_\beta ]$.

Now assume $\exists \sigma \forall \delta \in 2^\omega [\delta \in _I \sigma \rightarrow \delta \in \mathcal {G}_\beta ]$. We have to prove that $\exists \sigma \forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \gamma \in \mathcal {G}_\alpha ]$. First find $\sigma$ such that $\forall \delta \in 2^\omega [\delta \in _I \sigma \rightarrow \delta \in \mathcal {G}_\beta ]$. We will define $\sigma ^*$ as a strategy for player I in $(\omega \times 2)^\omega$ such that, for each c in Halfbin, if $c\in _I \sigma ^*$, then either $\varvec{\pi _{bin}}(c)\in _I\sigma$ or $\exists e\sqsubset c[e\in D_\alpha ]$.

We first define $\sigma ^*(\langle \;\rangle )$. Define $\delta$ in $2^\omega$ such that $\delta \in _I \sigma$ and $\forall i[\delta (2i+1)=0]$. Find m such that $\beta ({\overline{\delta }} m)\ne 0$ and distinguish two cases.

Case (a). $\exists n[2n<m\;\wedge \;\delta (2n)=1]$. Define $n_0:=\mu n[\delta (2n)=1]$ and $\sigma ^*(\langle \;\rangle ):=n_0$. Note that $\langle n_0\rangle \in _I\sigma ^*$ and $\varvec{\pi _{bin}}(\langle n_0\rangle )=\underline{{\overline{0}}}(2n_0)*\langle 1 \rangle ={\overline{\delta }}(2n_0+1)\in _I\sigma$.

Case (b). $\forall n[2n<m \rightarrow \delta (2n)=0]$. Conclude that $\varvec{\rho _{bin}}({\overline{\delta }} m)=\langle \;\rangle$ and $\beta (\langle \;\rangle )\ne 0$ and also $\alpha (\langle \;\rangle )\ne 0$. Define $\sigma ^*(\langle \;\rangle ):=0$. Note that $\langle 0\rangle \in _I \sigma ^*$ and $\exists e \sqsubset \langle 0\rangle [e\in D_\alpha ]$.

Now suppose $k>0$. Let c in $(\omega \times 2)^k$ be given such that $c\in _I\sigma ^*$. We have to define $\sigma ^*(c)$ and distinguish two cases.

Case 1. $\exists e\sqsubset c[e\in D_\alpha ]$. We then define $\sigma ^*(c):=0$. Note that $\exists e\sqsubset c*\langle 0\rangle [e\in D_\alpha ]$.

Case 2. $\lnot \exists e\sqsubset c[e\in D_\alpha ]$. Then $\varvec{\pi _{bin}}(c)\in _I \sigma$. Note that $length\bigl (\varvec{\pi _{bin}}(c)\bigr )$ is even and find l such that $2l:=length\bigl (\varvec{\pi _{bin}}(c)\bigr )$. Define $\delta$ in $2^\omega$ such that $\delta \in _I \sigma$ and $\varvec{\pi _{bin}}(c)\sqsubset \delta$ and $\forall i[2i+1>2\,l\rightarrow \delta (2i+1)=0]$. Find m such that $\beta ({\overline{\delta }} m)\ne 0$ and distinguish two cases.

Case (2a). $\exists n[2l\le 2n<m\;\wedge \;\delta (2n)=1]$. Define $n_0:=\mu n[2l\le 2n<m\;\wedge \;\delta (2n)=1]$ and $\sigma ^*(c):=n_0-l$. Note that $c*\langle n_0-l\rangle \in _I\sigma ^*$ and $\varvec{\pi _{bin}}(c*\langle n_0-l\rangle )=\varvec{\pi _{bin}}(c)*\underline{{\overline{0}}}(2n_0-2\,l)*\langle 1 \rangle ={\overline{\delta }}(2n_0+1)\in _I\sigma$.

Case (2b). $\forall n[2n<m \rightarrow \delta (2n)=0]$. Conclude: $\varvec{\rho _{bin}}({\overline{\delta }} m)=c$ and $\beta ({\overline{\delta }} m)\ne 0$ and $\alpha (c)=\beta ({\overline{\delta }} m)\ne 0$. Define $\sigma ^*(\langle \;\rangle ):=0$. Note that $c*\langle 0\rangle \in _I \sigma ^*$ and $\exists e \sqsubset c*\langle 0\rangle [e\in D_\alpha ]$.

This completes the definition of $\sigma ^*$.

Now assume $\gamma \in (\omega \times 2)^\omega$ and $\gamma \in _I \sigma ^*$. Find $\delta \in 2^\omega$ such that $\forall n[\varvec{\pi _{bin}}({\overline{\gamma }} n)\sqsubset \delta ]$. Find $\varepsilon$ in $2^\omega$ such that such that $\varepsilon \in _I\sigma$ and, for each n, if ${\overline{\delta }} n \in _I\sigma$, then ${\overline{\varepsilon }} n ={\overline{\delta }} n$. Find m such that ${\overline{\varepsilon }} m\in D_\beta$ and distinguish two cases.

Case $(*)$. ${\overline{\delta }} m ={\overline{\varepsilon }} m$. Conclude that ${\overline{\delta }} m \in D_\beta$ and $\varvec{\rho _{bin}}({\overline{\delta }} m)\in D_\alpha$ and $\gamma \in \mathcal {G}_\alpha$.

Case $(**)$. ${\overline{\delta }} m \ne {\overline{\varepsilon }} m$. Then ${\overline{\delta }} m \notin _I\sigma$ and $\varvec{\pi _{bin}}({\overline{\gamma }} m)\notin _I\sigma$ and $\exists e\sqsubset {\overline{\gamma }} m[e\in D_\alpha ]$. Conclude that $\gamma \in \mathcal {G}_\alpha$.

We thus see that $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I\sigma ^*\rightarrow \gamma \in \mathcal {G}_\alpha ]$.

Assume $\exists s\forall \delta \in 2^\omega [\delta \in _I s\rightarrow \delta \in \mathcal {G}_\beta ]$. We have to prove that $\exists s\forall \gamma \in (\omega \times 2)^\omega [\gamma \in s^*\rightarrow \gamma \in \mathcal {G}_\alpha ]$. First find s such that $\forall \delta \in 2^\omega [\delta \in _I s\rightarrow \delta \in \mathcal {G}_\beta ]$. Consider $p:=length(s)$. Find q such that, for all c in Halfbin, if $c\ge q$, then $\varvec{\pi _{bin}}(c)\ge p$. Define $s^*$ such that $length(s^*)=q$, inductively. For each $c<q$ in $\bigcup _k(\omega \times 2)^k$ such that $c\in _Is^*$, $s^*(c)$ is defined just as, in the previous paragraph, where we were given $\sigma \in 2^\omega$, $\sigma ^*(c)$ was defined, for each c in $\bigcup (\omega \times 2)^k$ such that $c\in _I\sigma ^*$. One then may prove that $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in s^*\rightarrow \gamma \in \mathcal {G}_\alpha ]$. $\square$

Lemma 9.8

One may prove the following statements in $\textsf{BIM}$.

• (i) $\varvec{\Sigma }^0_1$-$\textit{Det}_{\omega \times 2}^I \rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}}$ and $\lnot ! ( \varvec{\Sigma }^0_1$-$\overleftarrow{{\textbf {AC}}_{\omega ,2}})\rightarrow \lnot !( \varvec{\Sigma }^0_1$-$\textit{Det}_{\omega \times 2}^I)$.

• (ii) $\varvec{\Delta }^0_1$-$\textit{Det}^I_{\omega \times 2 \times \omega } \rightarrow \varvec{\Sigma }^0_1$-$\textit{Det}_{\omega \times 2}^I$ and $\lnot !\bigl (\varvec{\Sigma }^0_1$-$\textit{Det}_{\omega \times 2}^I\bigr ) \rightarrow \lnot !\bigl (\varvec{\Delta }^0_1$-$\textit{Det}^I_{\omega \times 2 \times \omega }\bigr )$.

• (iii) $\forall m[\varvec{\Delta }^0_1$-$\textit{Det}^I_{(\omega \times 2)^m}] \rightarrow \varvec{\Delta }^0_1$-$\textit{Det}^I_{\omega \times 2 \times \omega }$ and $\lnot !(\varvec{\Delta }^0_1$-$\textit{Det}_{\omega \times 2\times \omega }^I\bigr ) \rightarrow \exists m[\lnot !(\varvec{\Delta }^0_1$-$\textit{Det}^I_{(\omega \times 2)^m})]$.

• (iv) $\forall m[\varvec{\Sigma }^0_1$-$\textit{Det}^I_{(\omega \times 2)^\omega } \rightarrow \varvec{\Delta }^0_1$-$\textit{Det}^I_{(\omega \times 2)^m} \;\wedge \;$ $\lnot !(\varvec{\Delta }^0_1$-$\textit{Det}^I_{(\omega \times 2)^m}) \rightarrow \lnot !(\varvec{\Sigma }^0_1$-$\textit{Det}^I_{(\omega \times 2)^\omega })]$.

• (v) $\varvec{\Sigma }^0_1$-$\textit{Det}^I_{2^\omega } \rightarrow \varvec{\Sigma }^0_1$-$\textit{Det}^I_{(\omega \times 2)^\omega }$ and $\lnot !(\varvec{\Sigma }^0_1$-$\textit{Det}^I_{(\omega \times 2)^\omega }) \rightarrow \lnot !(\varvec{\Sigma }^0_1$-$\textit{Det}^I_{2^\omega })$, and $\varvec{\Sigma }^0_1$-$^*\textit{Det}^I_{2^\omega } \rightarrow \varvec{\Sigma }^0_1$-$^*\textit{Det}^I_{(\omega \times 2)^\omega }$ and $\lnot !(\varvec{\Sigma }^0_1$-$^*\textit{Det}^I_{(\omega \times 2)^\omega }) \rightarrow \lnot !(\varvec{\Sigma }^0_1$-$^*\textit{Det}^I_{2^\omega })$.

• (vi) $\varvec{\Sigma }^0_1$-$\textit{Det}^{II}_{2^\omega } \rightarrow \varvec{\Sigma }^0_1$-$\textit{Det}^I_{2^\omega }$ and $\lnot !(\varvec{\Sigma }^0_1$-$\textit{Det}^I_{2^\omega }) \rightarrow \lnot !(\varvec{\Sigma }^0_1$-$\textit{Det}^{II}_{2^\omega })$, and $\varvec{\Sigma }^0_1$-$^*\textit{Det}^{II}_{2^\omega } \rightarrow \varvec{\Sigma }^0_1$-$^*\textit{Det}^I_{2^\omega }$ and $\lnot !(\varvec{\Sigma }^0_1$-$^*\textit{Det}^I_{2^\omega }) \rightarrow \lnot !(\varvec{\Sigma }^0_1$-$^*\textit{Det}^{II}_{2^\omega })$.

• (vii) $\textbf{FT} \rightarrow \varvec{\Sigma }^0_1$-$^*\textit{Det}^{II}_{2^\omega }$ and $\lnot !(\varvec{\Sigma }^0_1$-$^*\textit{ Det}^{II}_{2^\omega }) \rightarrow \lnot !\textbf{FT}$.

Proof

(i) We prove: given any $\alpha$, one may construct $\beta$ such that

$$\begin{aligned}{} & {} \forall \gamma \in 2^\omega \exists n[\bigl (n,\gamma (n)\bigr ) \in E_{\alpha }]\rightarrow \forall \tau \in 2^\omega \exists n [\langle n, \tau (n)\rangle \in E_\beta ]\;\text {and}\\{} & {} \exists n[\langle n, 0\rangle \in E_\beta \; \wedge \; \langle n, 1\rangle \in E_\beta ]\rightarrow \exists n[(n,0) \in E_\alpha \;\wedge \;(n,1)\in E_\alpha ]. \end{aligned}$$

The two promised conclusions then follow easily.

Given $\alpha$, define $\beta$ such that $\forall n\forall i<2[\alpha (p)=(n,i)+1\leftrightarrow \beta (p) = \langle n, i\rangle +1]$ and $\forall p[\lnot \exists n\exists i<2[\alpha (p)=(n,i)+1]\rightarrow \beta (p) =0]$.

Clearly, $\beta$ satisfies the requirements.

(ii) We prove: given any $\alpha$, one may construct $\beta$ such that

$$\begin{aligned}{} & {} \forall \tau \in 2^\omega \exists n[\langle n, \gamma (n)\rangle \in E_\alpha ] \rightarrow \forall \tau \in 2^\omega \exists n\exists p[\langle n, \tau (n),p\rangle \in D_\beta ]\;\text {and}\\{} & {} \exists n\forall i<2\exists p[\langle n,i,p\rangle \in D_\beta ] \rightarrow \exists n\forall i<2[\langle n,i\rangle \in E_\alpha ]. \end{aligned}$$

The two promised conclusions then follow easily.

Given $\alpha$, define $\beta$ such that $\forall n\forall i<2\forall p[ \beta (\langle n,i,p\rangle ) \ne 0\leftrightarrow \alpha (p)= \langle n,i\rangle +1].$ Note that $\forall n\forall i<2[ \langle n, i\rangle \in E_\alpha \leftrightarrow \exists p[\langle n, i, p \rangle \in D_{\beta }]]$.

Clearly, $\beta$ satisfies the requirements.

(iii) Note that $\textsf{BIM}$ proves $\varvec{\Delta }^0_1$-$\textit{Det}^I_{(\omega \times 2)^2} \rightarrow \varvec{\Delta }^0_1$-$\textit{Det}^I_{\omega \times 2 \times \omega }$ and

$\lnot !(\varvec{\Delta }^0_1$-$\textit{Det}_{\omega \times 2\times \omega }^I\bigr ) \rightarrow \lnot !(\varvec{\Delta }^0_1$-$\textit{Det}^I_{(\omega \times 2)^2})$.

(iv) Let m be given. We prove: given any $\alpha$ one may construct $\beta$ such that

$$\begin{aligned}{} & {} \forall \tau \in 2^\omega \exists c\in (\omega \times 2)^m[ c \in _{II} \tau \; \wedge \; c \in D_\alpha ]\rightarrow \forall \tau \in 2^\omega \exists \gamma \in \mathcal (\omega \times 2)^\omega [\gamma \in _{II}\tau \;\wedge \; \gamma \in \mathcal {G}_\beta ]\\{} & {} \text {and} \;\exists \sigma \forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \gamma \in \mathcal {G}_\beta ]\rightarrow \exists s\forall c\in (\omega \times 2)^m[ c \in _I s \rightarrow c \in D_\alpha ]. \end{aligned}$$

The two promised conclusions then follow easily.

Given $\alpha$, define $\beta$ such that $\forall s [\beta (s)\ne 0\leftrightarrow \bigl ( s\in (\omega \times 2)^{m}\;\wedge \;\alpha (s)\ne 0\bigr )]$.

Observe that, if $\forall \tau \in 2^\omega \exists c\in (\omega \times 2)^m[ c \in _{II} \tau \; \wedge c \in D_\alpha ]$, then

$\forall \tau \in 2^\omega \exists \gamma \in 2^\omega [\gamma \in _{II}\tau \;\wedge \;\beta \bigl ({\overline{\gamma }}(2\,m)\bigr ) \ne 0 ]$, i.e. $\forall \tau \in 2^\omega \exists \gamma \in 2^\omega [ \gamma \in _{II}\tau \;\wedge \;\gamma \in \mathcal {G}_\beta ]$.

Let $\sigma$ in $2^\omega$ be given such that $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \exists n[\beta ({\overline{\gamma }} n) \ne 0]]$. Conclude that $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \alpha \bigl ( {\overline{\delta }}(2\,m)\bigr )\ne 0]$. Find N such that $\forall c \in \bigcup _{n\le 2\,m}\omega ^n[ c \in _I \gamma \rightarrow c <N]$ and define $s:= {\overline{\sigma }} N$. Conclude that $\forall c\in (\omega \times 2)^m[ c \in _I s \rightarrow c \in D_\alpha ]$.

We thus see that $\beta$ satisfies the requirements.

(v) For each $\alpha$, one may construct $\beta$ such that

$$\begin{aligned}{} & {} \forall \tau \in 2^\omega \exists \gamma \in (\omega \times 2)^\omega [\gamma \in _{II}\tau \;\wedge \;\gamma \in \mathcal {G}_\alpha ] \rightarrow \\{} & {} \forall \tau \in 2^\omega \exists \delta \in 2^\omega [\delta \in _{II}\tau \;\wedge \;\delta \in \mathcal {G}_\beta ]\;\text {and}\\{} & {} \exists \sigma \forall \delta \in 2^\omega [\delta \in _I \sigma \rightarrow \delta \in \mathcal {G}_\beta ]\rightarrow \exists \sigma \forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \gamma \in \mathcal {G}_\alpha ]\;\text {and}\\{} & {} \exists s\forall \delta \in 2^\omega [\delta \in _I s\rightarrow \delta \in \mathcal {G}_\beta ]\rightarrow \exists s\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I s\rightarrow \gamma \in \mathcal {G}_\alpha ]. \end{aligned}$$

The proof has been given in Lemma 9.7. The promised conclusions follow easily.

(vi) We prove that, given any $\alpha$, one may construct $\beta$ such that

$$\begin{aligned}{} & {} \forall \sigma \in 2^\omega \exists c \in 2^{<\omega }[c \in _{I} \sigma \;\wedge \; \alpha (c) \ne 0] \rightarrow \forall \tau \in 2^\omega \exists d \in 2^{<\omega }[d \in _{II} \tau \;\wedge \; \beta (d) \ne 0]\;\text {and}\\{} & {} \exists \tau \in 2^\omega \forall \delta \in 2^\omega [\delta \in _{II} \tau \rightarrow \exists n[\beta ({\overline{\delta }} n) \ne 0]]\rightarrow \exists \sigma \in 2^\omega \forall \delta \in 2^\omega [\delta \in _{I} \sigma \rightarrow \exists n[\alpha ({\overline{\delta }} n) \ne 0]] \\{} & {} \text {and}\; \exists t\forall \delta \in 2^\omega [\delta \in _{II} t\rightarrow \exists n[\beta ({\overline{\delta }} n) \ne 0]]\rightarrow \exists t\forall \delta \in 2^\omega [\delta \in _{I} t\rightarrow \exists n[\alpha ({\overline{\delta }} n) \ne 0]]. \end{aligned}$$

The promised conclusions then follow easily.

Given $\alpha$, define $\beta$ such that $\beta (0) =0$ and $\forall c \in 2^{<\omega }[\beta (\langle 0 \rangle *c) = \beta (\langle 1 \rangle *c) = \alpha (c)]$.

Assume that $\forall \sigma \in 2^\omega \exists c \in 2^{<\omega }[c \in _{I} \sigma \;\wedge \; \alpha (c) \ne 0]$. Let $\tau$ in $2^\omega$ be given. Define $\sigma$ such that $\forall c \in 2^{<\omega }[\sigma (c) = \tau \bigl (\langle 0\rangle *c\bigr )]$. Find c in $2^{<\omega }$ such that $c \in _{I} \sigma$ and $\alpha (c) \ne 0$.Define $d:=\langle 0\rangle *c$ and note $d \in _{II} \tau$ and $\beta (d) \ne 0$. Conclude that $\forall \tau \in 2^\omega \exists d \in 2^{<\omega }[d \in _{II} \tau \;\wedge \; \beta (d) \ne 0]$.

Let $\tau$ in $2^\omega$ be given such that $\forall \delta \in 2^\omega [\delta \in _{II} \tau \rightarrow \exists n[\beta ({\overline{\delta }} n) \ne 0]]$. Define $\sigma$ such that $\forall c\in 2^{<\omega }\forall i<2[\sigma (\langle i\rangle *c) = \tau (c)]$. Note that $\forall \delta \in 2^\omega [\delta \in _{I} \sigma \rightarrow \langle 0 \rangle *\delta \in _{II} \tau ]$, so $\forall \delta \in 2^\omega [\delta \in _{I} \sigma \rightarrow \exists n[\beta ( \overline{\langle 0\rangle *\delta } n) \ne 0]]$ and $\forall \delta \in 2^\omega [\delta \in _{I} \sigma \rightarrow \exists n[\alpha ({\overline{\delta }} n)\ne 0]]$.

Let t be given such that $\forall \delta \in 2^\omega [\delta \in _{II} t\rightarrow \exists n[\beta ({\overline{\delta }} n) \ne 0]]$. Define s such that, $\forall c\in 2^{<\omega }[\langle 0\rangle *c<length(t)\rightarrow s(c) = t(\langle 0 \rangle *c)]$. Conclude, as above, that $\forall \delta \in 2^\omega [\delta \in _{I} s\rightarrow \exists n[\alpha ({\overline{\delta }} n)\ne 0]]$.

We thus see that $\beta$ satisfies the requirements.

(vii) We prove that, for each $\alpha$, there exists $\beta$ such that

$$\begin{aligned}{} & {} \forall \gamma \in 2^\omega \exists s[s \in _I \gamma \;\wedge \; \alpha (s) \ne 0]\rightarrow \textit{Bar}_{2^\omega }(D_\beta )\;\text {and}\\{} & {} \exists m[Bar_{2^\omega }(D_{{\overline{\beta }} m})]\rightarrow \exists c\forall \delta \in 2^\omega [ \delta \in _{II} c \rightarrow \exists n[\alpha ({\overline{\delta }} n) \ne 0]]. \end{aligned}$$

The two promised conclusions then follow easily.

Given $\alpha$, define $\beta$ such that $\forall c \in 2^{<\omega }[\beta (c) \ne 0\leftrightarrow \exists s<c[s \in _{I} c\;\wedge \;\alpha (s) \ne 0]]$.

Assume $\forall \gamma \in 2^\omega \exists s[s \in _I \gamma \;\wedge \; \alpha (s) \ne 0]$. Clearly, $\forall \gamma \in 2^\omega \exists n[\beta ({\overline{\gamma }} n) \ne 0]$, i.e. $\textit{Bar}_{2^\omega }(D_\beta )$.

Let m be given such that $Bar_{2^\omega }(D_{{\overline{\beta }} m})$. Define $X:=\{s\in 2^{<\omega }\mid length(s)=m\;\wedge \;\exists n \le m[\alpha ({\overline{s}} n) \ne 0]\}$. Note that $\forall b\exists s[s \in _I b \;\wedge \; s \in X]$. According to Theorem 9.3, $\textit{Det}^{II}_{2^{<\omega }\cap \omega ^m}(X)$. Find c such that $\forall s\in 2^{<\omega }[\bigl (length(s)=m\;\wedge \; s \in _{II} c\bigr ) \rightarrow s \in X]$. Conclude that $\forall \delta \in 2^\omega [\delta \in _{II} c \rightarrow {\overline{\delta }} m \in X]$ and $\forall \delta \in 2^\omega [ \delta \in _{II} c \rightarrow \exists n[\alpha ({\overline{\delta }} n) \ne 0]]$.

We thus see that $\beta$ satisfies the requirements. $\square$

Theorem 9.9

• (i) $\textsf{BIM} \vdash \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega , 2}} \leftrightarrow \varvec{\Sigma }^0_1$-$\textit{Det}_{\omega \times 2}^I \leftrightarrow \varvec{\Delta }^0_1$-$\textit{Det}^I_{\omega \times 2 \times \omega } \leftrightarrow \forall m[\varvec{\Delta }^0_1$-$\textit{Det}^{I}_{(\omega \times 2)^m}] \leftrightarrow \varvec{\Sigma }^0_1$-$\textit{Det}^I_{(\omega \times 2)^\omega }\leftrightarrow \varvec{\Sigma }^0_1$-$\;^*\textit{ Det}^I_{(\omega \times 2)^\omega }\leftrightarrow \varvec{\Sigma }^0_1$-$\textit{Det}^I_{2^\omega }\leftrightarrow \varvec{\Sigma }^0_1$-$\;^*\textit{ Det}^I_{2^\omega }\leftrightarrow \varvec{\Sigma }^0_1$-$\textit{Det}^{II}_{2^\omega }\leftrightarrow \varvec{\Sigma }^0_1$-$\;^*\textit{ Det}^{II}_{2^\omega }\leftrightarrow \textbf{FT}$.

• (ii) $\textsf{BIM} \vdash \lnot !(\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega , 2}}) \leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\textit{Det}_{\omega \times 2}^I) \leftrightarrow \lnot !(\varvec{\Delta }^0_1$-$\textit{Det}^I_{\omega \times 2 \times \omega }) \leftrightarrow \exists m[\lnot !(\varvec{\Delta }^0_1$-$\textit{Det}^{I}_{(\omega \times 2)^m})] \leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\textit{Det}^I_{(\omega \times 2)^\omega })\leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\;^*\textit{ Det}^I_{(\omega \times 2)^\omega })\leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\textit{Det}^I_{2^\omega })\leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\;^*\textit{ Det}^I_{2^\omega })\leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\textit{Det}^{II}_{2^\omega })\leftrightarrow \lnot !(\varvec{\Sigma }^0_1$-$\;^*\textit{ Det}^{II}_{2^\omega })\leftrightarrow \lnot !\textbf{FT}$.

Proof

Use Lemmas 5.3 and 9.8. $\square$

9.5 The intuitionistic determinacy theorem

Recall that $\mathcal {X}\subseteq (\omega \times 2)^\omega$ is I-determinate in $(\omega \times 2)^\omega$ if and only if

$$\begin{aligned} \forall \tau \in 2^\omega \exists \gamma [\gamma \in _{II}\tau \;\wedge \; \gamma \in \mathcal {X}]\rightarrow \exists \sigma \forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \gamma \in \mathcal {X}]. \end{aligned}$$

We now define: $\mathcal {X}\subseteq (\omega \times 2)^\omega$ is weakly I-determinate in $(\omega \times 2)^\omega$ if and only if

$$\begin{aligned}{} & {} \forall \varphi :2^\omega \rightarrow (\omega \times 2)^\omega [\forall \tau \in 2^\omega [\varphi |\tau \in _{II}\tau \;\wedge \; \varphi |\tau \in \mathcal {X}]\rightarrow \\{} & {} \exists \sigma \forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \gamma \in \mathcal {X}]]. \end{aligned}$$

A (continuous) function $\varphi :2^\omega \rightarrow (\omega \times 2)^\omega$ such that $\forall \tau \in 2^\omega [\varphi |\tau \in _{II} \tau ]$ will be called an anti-strategy for player I in $(\omega \times 2)^\omega$. Note that the Second Axiom of Continuous Choice,¹⁸${\textbf {AC}}_{\omega ^\omega ,\omega ^\omega }={\textbf {AC}}_{1,1}$, implies, for every subset $\mathcal {X}\subseteq (\omega \times 2)^\omega$: if $\forall \tau \in 2^\omega \exists \gamma \in (\omega \times 2)^\omega [\gamma \in _{II}\tau \;\wedge \;\gamma \in \mathcal {X}]$, then $\exists \varphi : 2^\omega \rightarrow (\omega \times 2)^\omega \forall \tau \in 2^\omega [\varphi |\tau \in _{II}\tau \;\wedge \;\varphi |\tau \in \mathcal {X}]$. ${\textbf {AC}}_{\omega ^\omega ,\omega ^\omega }$ thus implies that, if $\mathcal {X}\subseteq (\omega \times 2)^\omega$ is weakly I-determinate in $(\omega \times 2 )^\omega$, then $\mathcal {X}$ is I-determinate in $(\omega \times 2 )^\omega$.

Earlier versions of the next Theorem may be found in [32, Chapter 16] and [42].

Theorem 9.10

(Intuitionistic Determinacy Theorem) The following statements are equivalent in $\textsf{BIM}$:

• (i) $\textbf{FT}$.

• (ii) For every anti-strategy $\varphi$ for player I in $(\omega \times 2)^\omega$ there exists a strategy $\sigma$ for player I in $(\omega \times 2)^\omega$ such that $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I \sigma \rightarrow \exists \tau \in 2^\omega [\gamma =\varphi |\tau ]].$

Proof

(i) $\Rightarrow$ (ii). Let $\varphi :2^\omega \rightarrow (\omega \times 2)^\omega$ be given such that $\forall \tau \in 2^\omega [\varphi |\tau \in _{II} \tau ]$. We intend to develop a strategy $\sigma$ for player I in $(\omega \times 2)^\omega$ such that Player I, whenever he obeys $\sigma$ and develops, together with player II, $\gamma$ in $(\omega \times 2)^\omega$, will be able to construct, simultaneously, a strategy $\tau$ for player II in $(\omega \times 2 )^\omega$ such that $\gamma =\varphi |\tau$. The infinite sequence $\gamma$ must be the answer given by the anti-strategy $\varphi$ to player II’s strategy $\tau$. So, while playing $\gamma$, player I conjectures a strategy $\tau$ that player II may be assumed to follow during this very play.

Let c, t be given such that $c\in \bigcup _k(\omega \times 2)^k$ and $t\in 2^{<\omega }$. We define: with respect to the given anti-strategy $\varphi$, t is, at the position c, a safe (partial) conjecture by player I about the strategy followed by player II, notation: $\textit{Safe}( c, t)$, if and only if¹⁹$\forall \rho \in 2^\omega \exists u\in 2^{<\omega }[t\sqsubset u\;\wedge \; ^cu \sqsubset \rho \;\wedge \; c \sqsubseteq \varphi |u].$ Note²⁰ that $\textit{Safe}( c, t)\leftrightarrow \forall \rho \in 2^\omega \exists \tau \in 2^\omega [t\sqsubset \tau \;\wedge \; ^c\tau = \rho \;\wedge \; c \sqsubset \varphi |\tau ]$. If $\textit{Safe}( c, t)$, then player I, at the position c, may extend every strategy $\rho$ of player II ‘above c’, that is, on positions d in $(\omega \times 2)^{<\omega }\times \omega$ such that $c\sqsubseteq d$, to a strategy $\tau$ of player II on the whole of $(\omega \times 2)^{<\omega }\times \omega$ such that $t\sqsubset \tau$ and $c\sqsubset \varphi |\tau$. Note that $\textit{Safe}(\langle \;\rangle , \langle \;\rangle )$.

We now prove that one may decide, given c in $\bigcup _k (\omega \times 2)^k$ and t in $2^{<\omega }$, if $\textit{Safe}(c,t)$ holds or not, i.e. we prove that $\forall c\in \bigcup _k(\omega \times 2)^k\forall t\in 2^{<\omega }[\textit{Safe}( c, t) \vee \lnot \textit{Safe}( c, t)].$ Let c in $\bigcup _k(\omega \times 2)^k$ and t in $2^{<\omega }$ be given. Define $\beta$ such that $\forall u\in 2^{<\omega } [\beta (u)\ne 0\leftrightarrow (c\sqsubseteq \varphi |u\;\vee \;c\perp \varphi |u)]$. Note that $Bar_{2^\omega }(D_\beta )$, and that $\forall u[u\in D_\beta \rightarrow \forall i<2[u*\langle i \rangle \in D_\beta ]]$. Using ${\textbf {FT}}$, find $n>\textit{length}(t)$ such that $\forall u\in 2^{<\omega }[length(u)=n \rightarrow u \in D_\beta ]$. Note that $\forall u[\textit{length}(^cu)\le length(u)\le n]$ and $\textit{Safe}(c,t)\leftrightarrow \forall r \in 2^{<\omega }[length(r)=n \rightarrow \exists u \in 2^{<\omega }[ t\sqsubseteq u \;\wedge \; ^cu \sqsubseteq r\;\wedge \; c\sqsubseteq \varphi |u].$ We thus see that one may indeed decide $\textit{Safe}(c,t)\;\vee \;\lnot \textit{Safe}(c,t)$.

We want to define a function taking these decisions, i.e. we want to define $\nu$ such that $\forall c\in \bigcup _k(\omega \times 2)^k\forall t\in 2^{<\omega }[\nu (c,t)\ne 0\leftrightarrow \textit{Safe}( c,t )].$ We first define $\psi$ such that, for all c in $\bigcup _k(\omega \times 2)^k$, for all t in $2^{<\omega }$, $\psi (c,t):=\mu n[n>\textit{length}(t)\;\wedge \;\forall u \in 2^{<\omega }[length(u)=n\rightarrow c\sqsubseteq \varphi | u \;\vee \;c\perp \varphi |u]],$ and then define $\nu$ such that, for all c in $\bigcup _k(\omega \times 2)^k$, for all t in $2^{<\omega }$, $\nu (c,t)\ne 0\leftrightarrow$ $\forall r \in 2^{<\omega }[length(r)=\psi (c,t) \rightarrow \exists u \in 2^{<\omega }[length(u)={\psi (c,t)}\;\wedge \; t\sqsubseteq u \;\wedge \; ^cu \sqsubseteq r\;\wedge \; c\sqsubseteq \varphi |u]].$ We also define $\chi$ such that, for all c in $\bigcup _k(\omega \times 2)^k$, for all t in $2^{<\omega }$, if $\nu (c,t)=0$, then $\chi (c,t)=\mu r[r\in 2^{<\omega } \;\wedge \; length(r)=\psi (c,t)\;\wedge \;\lnot \exists u \in 2^{<\omega }[length(u)={\psi (c,t)}\;\wedge \; t \sqsubseteq u\;\wedge \; ^cu \sqsubseteq r\;\wedge \; c\sqsubseteq \varphi |u]].$ Note that, if $\nu ( c, t)=0$, then $\forall \rho \in 2^\omega [\chi (c,t)\sqsubset \rho \rightarrow \lnot \exists u\in 2^{<\omega }[ t\sqsubseteq u\;\wedge \;^cu\sqsubset \rho \;\wedge \; c\sqsubseteq \varphi |u]]$.

Let n, c, t be given such that $c\in (\omega \times 2)^n$ and t in $2^{<\omega }$ and $\textit{Safe}( c, t)$. We shall prove that $\forall \rho \in 2^\omega \exists d\in \bigcup _{k>0}(\omega \times 2)^k[d\in _{II}\rho \;\wedge \; \exists i<2[\textit{Safe}( c*d, t*\langle i \rangle )]].$ Let $\rho$ in $2^\omega$ be given. Find u in $2^{<\omega }$ such that $t\sqsubset u\;\wedge \;^cu\sqsubset \rho \;\wedge \; c\sqsubseteq \varphi |u$. Find $i<2$ such that $t*\langle i\rangle \sqsubseteq u$. Find p such that, for all d in $\bigcup _k(\omega \times 2)^k\times \omega$, if $d< length(^cu)$, then $length(d)<2p$. Define $D:=\{d\in (\omega \times 2)^p\mid d \in _{II}\;\rho \;\wedge \;\exists \tau \in 2^\omega [c*d\sqsubset \varphi |\tau ]\}$. It follows from ${\textbf {FT}}$ that $E:=\{\overline{\varphi |\tau }(2n+2p)\mid \tau \in 2^\omega \}$ is a finite²¹ subset of $\omega$. Note that, for all d in $(\omega \times 2)^p$, $d\in D$ if and only if $\exists b \in E[c*d = b]$ and conclude that also D is a finite subset of $\omega$. Assume $\forall d\in D[\lnot \textit{Safe}(c*d, t*\langle i\rangle )]$. Define $\rho ^*$ in $2^\omega$ such that for all v in $\bigcup _{k<n+p} (\omega \times 2)^k\times \omega [\rho ^*(v)= \rho (v)]$ (and, therefore, $^cu\sqsubset \rho ^*$) and $\forall d \in D[d\in _{II}\rho ^*]$ and $\forall d\in D[\chi (c*d, t*\langle i\rangle )\sqsubset \; ^{d}(\rho ^*)]$. Find $\tau$ in $2^\omega$ such that $u\sqsubset \tau$ and $^c\tau =\rho ^*$ and consider $\varphi |\tau$. Note that $c\sqsubseteq \varphi |u\sqsubset \varphi |\tau$. Find d in D such that $c*d\sqsubset \varphi |\tau$. Note that $^{c*{d}}\tau =\;^{d}(\rho ^*)$ and $\chi (c*d, t*\langle i\rangle )\sqsubset \;^d(\rho ^*)$. Conclude that $\lnot \exists v\in 2^{<\omega }[t*\langle i\rangle \sqsubseteq v \;\wedge \; ^{c*d}v \sqsubseteq \;^d(\rho ^*)\;\wedge \; c*d \sqsubseteq \varphi |v]$. On the other hand, $t*\langle i\rangle \sqsubseteq u\sqsubset \tau$. Find m such that $c*d\sqsubseteq \varphi |{\overline{\tau }} m$. Note that $^c\tau =\rho ^*$ and, therefore, $^{c*d}{\overline{\tau }} m \sqsubset \;^d(\rho ^*)$. Conclude that $t*\langle i\rangle \sqsubset {\overline{\tau }} m\;\wedge \;^{c*d}{\overline{\tau }} m \sqsubset \;^d(\rho ^*)\;\wedge \;c*d\sqsubseteq \varphi |{\overline{\tau }} m$. Defining $v:={\overline{\tau }} m$, we see that we obtain a contradiction. We conclude that $\lnot \forall d\in D[\lnot \textit{Safe}(c*d, t*\langle i\rangle )]$. As D is finite, conclude that $\exists d\in D[\textit{Safe}( c*d, t*\langle i \rangle )]$. We thus see that, for all c in $\bigcup _k (\omega \times 2)^k$, for all t in $2^{<\omega }$,

$$\begin{aligned} \textit{Safe}(c,t)\rightarrow \forall \rho \in 2^\omega \exists d\in \bigcup _{k>0}(\omega \times 2)^k[d\in _{II}\rho \;\wedge \; \textit{Safe}( c*d, t*\langle i \rangle )]. \end{aligned}$$

This key fact enables us to develop the promised strategy for player I.

Let c, t be given such that $c \in \bigcup _k (\omega \times 2)^k$ and $t\in 2^{<\omega }$ and $\textit{Safe}(c,t)$. Then $\forall \rho \in 2^\omega \exists d\in \bigcup _{k>0}(\omega \times 2)^k[d\in _{II}\rho \;\wedge \; \textit{Safe}( c*d, t*\langle i \rangle )]$. Using $\varvec{\Sigma }^0_1$-$\textit{Det}^I_{(\omega \times 2)^\omega }$, an equivalent of ${\textbf {FT}}$, see Theorem  9.9(i), find $\upsilon$ such that $\forall \delta [\delta \in _I \upsilon \rightarrow \exists k[ \textit{Safe}( c*{\overline{\delta }} (2k+2), t*\langle i \rangle )]].$ Note that the set $\{\delta \in (\omega \times 2)^\omega \mid \delta \in _I\upsilon \}$ is an explicit fan. Using ${\textbf {FT}}$, find m such that $\forall \delta [\delta \in _I \upsilon \rightarrow \exists k \le m[\textit{Safe}( c*{\overline{\delta }} (2k+2), t*\langle i \rangle )]]$. Find p such that $\forall \delta [\delta \in _I\upsilon \rightarrow {\overline{\delta }}(2\,m+2)<p]$ and define $s:={\overline{\upsilon }} p$. Note that for all d in $(\omega \times 2)^{<\omega }$, if $d>\textit{length}(s)$ and $d\in _I s$, then $\exists n[0<2n\le \textit{length}(d) \;\wedge \; \textit{Safe}(c*{\overline{d}}(2n), t*\langle i \rangle )].$

Note that, for all s, the set of all d in $(\omega \times 2)^{<\omega }$ such that $d>\textit{length}(s)$ and $\forall n[2n<length(d) \rightarrow {\overline{d}}(2n)<length(s)]$ and $d\in _Is$ is a finite subset of $\omega$.

Define $\alpha$ such that for all c in $(\omega \times 2)^{<\omega }$, for all t in $2^{<\omega }$, for all s, $\alpha (c,t,s)\ne 0$ if and only if, for all d in $(\omega \times 2)^{<\omega }$, if $d>\textit{length}(s)$ and $\forall n[2n<length(d ) \rightarrow {\overline{d}}(2n)<length(s)]$ and $d\in _I s$, then $\exists n[0<2n\le \textit{length}(d) \;\wedge \;\exists i<2[ \textit{Safe}(c*{\overline{d}}(2n), t*\langle i \rangle )]].$ Conclude that $\forall c \in (\omega \times 2)^{<\omega }\forall t \in 2^{<\omega }[\textit{Safe}(c,t)\rightarrow \exists s[\alpha (c,t,s)\ne 0]]$. Define $\eta$ such that $\forall c \in (\omega \times 2)^{<\omega }\forall t\in 2^{<\omega }[\textit{Safe}( c, t)\rightarrow \eta (c,t)=\mu s[\alpha (c,t,s)\ne 0]].$ Define $\lambda$ such that $\lambda (\langle \; \rangle ) = \langle \; \rangle$, and, for all c in $(\omega \times 2)^{<\omega }$, for all $\langle p, i\rangle$ in $\omega \times 2$, if there exists $j<2$ such that $\textit{Safe}\bigl ( c*\langle p, i \rangle , \lambda (c)*\langle j \rangle \bigr )$, then $\lambda (c*\langle p, i \rangle ) = \lambda (c)*\langle j_0 \rangle$, where $j_0$ is the least such j, and, if not, then $\lambda (c*\langle p, i\rangle )= \lambda (c)$. Note that $\forall c \in (\omega \times 2)^{<\omega }[\textit{Safe}\bigl ( c, \lambda (c)\bigr )]$. Note that $\forall c\in (\omega \times 2)^{<\omega }\forall d\in (\omega \times 2)^{<\omega }[c\sqsubseteq d\rightarrow \lambda (c)\sqsubseteq \lambda (d)]$.

Now define a strategy $\sigma$ for player I in $(\omega \times 2)^\omega$ as follows. Let c in $(\omega \times 2)^{<\omega }$ be given. Find $n_0:=\mu n[\textit{Safe}( {\overline{c}}(2n), \lambda (c))]$. Find d such that $c = {\overline{c}}(2n_0)*d$. Define $\sigma (c):= \bigl (\eta ({\overline{c}}(2n_0), \lambda (c))\bigr )(d)$.

We now prove that $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I\sigma \rightarrow \exists \tau \in 2^\omega [\varphi |\tau =\gamma ]]$.

Let $\gamma$ be an element of $(\omega \times 2)^\omega$ such that $\gamma \in _I \sigma$. We first prove that $\forall n\exists m\ge n[\lambda \bigl ({\overline{\gamma }}(2\,m)\bigr ) \sqsubset \lambda \bigl ({\overline{\gamma }}(2\,m+2)\bigr )]$. Let n be given. Find $n_0:=\mu k[\textit{Safe}\bigl ({\overline{\gamma }}(2k), \lambda ({\overline{\gamma }} (2n)\bigr )]$. Define $c:={\overline{\gamma }} (2n_0)$ and $t:=\lambda (c)$. Consider $s:=\eta (c,t)$. Find $\delta$ such that $\delta \in _I s$ and $\forall k[\delta (2k+1)= \gamma (2n_0+2k+1)]$. Find $k_0:=\mu k[\exists i<2[\textit{Safe}\bigl (c*{\overline{\delta }}(2k+2), t*\langle i \rangle \bigr )]$. Note that $c*{\overline{\delta }} (2k_0+2) \sqsubset \gamma$. Note that $\lambda (c)=\lambda \bigl ({\overline{\gamma }} (2n)\bigr )= \lambda \bigl ({\overline{\gamma }} (2n_0+2k_0)\bigr )\sqsubset \lambda \bigl ({\overline{\gamma }}(2n_0+2k_0+2)\bigr )$. Defining $m:=n_0+k_0$, we see that $m\ge n$ and $\lambda \bigl ({\overline{\gamma }}(2\,m)\bigr ) \sqsubset \lambda \bigl ({\overline{\gamma }}(2\,m+2)\bigr )$. We thus see that $\forall n\exists m\ge n[\lambda \bigl ({\overline{\gamma }}(2\,m)\bigr ) \sqsubset \lambda \bigl ({\overline{\gamma }}(2\,m+2)\bigr )]$. Find $\tau$ in $2^\omega$ such that $\forall n[ \lambda \bigl ({\overline{\gamma }}(2n)\bigr )\sqsubset \tau ]$. Note that $\forall n \exists \zeta \in 2^\omega [ {\overline{\tau }} n\sqsubset \zeta \;\wedge \; {\overline{\gamma }}(2n)\sqsubset \varphi |\zeta ]$. Conclude that $\varphi |\tau = \gamma$. We thus see that $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I\sigma \rightarrow \exists \tau \in 2^\omega [\varphi |\tau =\gamma ]]$.

(ii) $\Rightarrow$ (i). Assume (ii). We shall prove $\varvec{\Sigma }^0_1$-$\textit{Det}^I_{\omega \times 2}$. Using Theorem 9.9(i), we then may conclude ${\textbf {FT}}$.

Let $\alpha$ be given such that $\forall \tau \in 2^\omega \exists n [\langle n, \tau (\langle n\rangle )\rangle \in E_\alpha ]$, i.e. $\forall \tau \in 2^\omega \exists n\exists p [\alpha (p)=\langle n, \tau (\langle n\rangle )\rangle +1]$, i.e. $\forall \tau \in 2^\omega \exists p [\alpha (p')=\langle p'', \tau (\langle p''\rangle )\rangle +1]$. Define $\psi :2^\omega \rightarrow \omega$ such that $\forall \tau \in 2^\omega [\psi (\tau )=\mu p[\alpha (p') = \langle p'',\tau (\langle p''\rangle )\rangle +1]$. Define $\varphi : 2^\omega \rightarrow (\omega \times 2)^\omega$ such that $\forall \tau \in 2^\omega [\varphi |\tau =\langle \psi ''(\tau ), \tau (\langle \psi ''(\tau )\rangle )\rangle *\underline{0}]$. Find $\sigma$ such that $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I\sigma \rightarrow \exists \tau \in 2^\omega [\gamma =\varphi |\tau ]]$, and, therefore, $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I\sigma \rightarrow \exists q[\alpha (q)={\overline{\gamma }} 2+1]]$ and $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I\sigma \rightarrow {\overline{\gamma }} 2 \in E_\alpha ]$. Define $n:=\sigma (\langle \;\rangle )$ and conclude that $\forall i<2[\langle n,i\rangle \in _I \sigma ]$ and $\forall i<2[\langle n, i\rangle \in E_\alpha ]$. We thus see that $\forall \alpha [\forall \tau \in 2^\omega \exists n[\langle n, \tau (\langle n\rangle )\rangle \in E_\alpha ]\rightarrow \exists n\forall i<2[\langle n, i\rangle \in E_\alpha ]]$, i.e. $\varvec{\Sigma }^0_1$-$Det^I_{\omega \times 2}$. $\square$

Corollary 9.11

$\textsf{BIM}+\textbf{FT}$ proves the following scheme:

Every $\mathcal {X}\subseteq (\omega \times 2)^\omega$ is weakly I-determinate.

The theorem and its corollary may be generalized. One may replace $(\omega \times 2)^\omega$ by any spread $\mathcal {F}$ satisfying the condition: $\exists \zeta \forall \alpha \in \mathcal {F} \forall n[\alpha (2n+1) \le \zeta (n)]$.

Let $\varphi :2^\omega \rightarrow (\omega \times 2)^\omega$ be an anti-strategy for player I in $(\omega \times 2)^\omega$. We define: $\varphi$ fails to translate into a strategy for player I if and only if

$$\begin{aligned}\lnot \exists \sigma \forall \gamma \in (\omega \times 2)^\omega [ \gamma \in _I \sigma \rightarrow \exists \tau \in 2^\omega [ \gamma = \varphi |\tau ]].\end{aligned}$$

Note that $\lnot !{\textbf {FT}}$ implies the existence of an anti-strategy for player I in $(\omega \times 2)^\omega$ that fails to translate into a strategy for player I in $(\omega \times 2)^\omega$. One argues as follows. Using $\lnot !{\textbf {FT}}$ and Theorem 9.9(ii), find $\alpha$ such that $\forall \tau \in 2^\omega \exists n[\langle n, \tau (\langle n\rangle )\rangle \in E_\alpha ]$ and $\lnot \exists n\forall i<2[\langle n, i\rangle \in E_\alpha ]$. As in the proof of Theorem 9.10(ii) $\Rightarrow$ (i), find an anti-strategy $\varphi$ for player I in $(\omega \times 2)^\omega$ such that $\forall \tau \in 2^\omega [\overline{(\varphi |\tau )}2 \in E_\alpha ]$. Assume $\sigma$ is a strategy for player I in $(\omega \times 2)^\omega$ such that $\forall \gamma \in (\omega \times 2)^\omega [\gamma \in _I\sigma \rightarrow \exists \tau \in 2^\omega [\gamma =\varphi |\tau ]]$. Consider $n:= \sigma (\langle \; \rangle )$ and conclude that $\forall i<2[\langle n, i \rangle \in E_\alpha ]$. Contradiction.

We did not find an argument proving $\lnot !{\textbf {FT}}$ from the assumption of the existence of an anti-strategy for player I in $(\omega \times 2)^\omega$ that fails to translate into a strategy for player I in $(\omega \times 2)^\omega$.

10 The (uniform) intermediate value theorem

10.1. The Intermediate Value Theorem, ${\textbf {IVT}}$²²

For all $\varphi$ in $\mathcal {R}^{[0,1]}$,

if $\exists \gamma \in [0,1]^2[\ \varphi ^{`\mathcal {R}}(\gamma ^{\upharpoonright 0} ) \le _\mathcal {R} 0_\mathcal {R} \le _\mathcal {R} \varphi ^{`\mathcal {R}}(\gamma ^{\upharpoonright 1})]$, then $\exists \gamma \in [0,1][\varphi ^{`\mathcal {R}}(\gamma ) =_\mathcal {R} 0_\mathcal {R}].$

${\textbf {IVT}}$ fails constructively. The next two theorems are similar to [3, Chapter 3, Theorem 2.4] and [28, Theorem 6(iv) and (iii)].

Theorem 10.1

$\textsf{BIM}\vdash \textbf{IVT} \rightarrow \textbf{LLPO}$.

Proof

Assume ${\textbf {IVT}}$. Let $\beta$ be given. Define $\delta$ in $\mathcal {R}$ such that, for each n, if $\underline{{\overline{0}}} n\sqsubset \beta$, then $\delta (n)=(-\frac{1}{2^n}, \frac{1}{2^n})$, and, if $\underline{{\overline{0}}}n \perp \beta$ and $p_0:=\mu p[\beta (p)\ne 0]$, then $\delta (n) =\bigl ((-1)^{p_0}\frac{1}{2^{p_0+1}} - \frac{1}{2^{n+3}}, (-1)^{p_0}\frac{1}{2^{p_0+1}} + \frac{1}{2^{n+3}}\bigr )$. Note that $\delta >_\mathcal {R} 0_\mathcal {R} \leftrightarrow \exists p[2p=\mu n[\beta (n)\ne 0]]$ and $\delta <_\mathcal {R} 0_\mathcal {R} \leftrightarrow \exists p[2p+1=\mu n[\beta (n)\ne 0]]$. Find $\varphi$ in $\mathcal {R}^{[0,1]}$ such that $\varphi ^{`\mathcal {R}}(0_\mathcal {R}) =_\mathcal {R} (-1)_\mathcal {R}$, and $\varphi ^{`\mathcal {R}}(\frac{1}{3}) =_\mathcal {R} \varphi ^{`\mathcal {R}}(\frac{2}{3}) =_\mathcal {R} \delta$ and $\varphi ^{`\mathcal {R}}(1_\mathcal {R}) = _\mathcal {R}1_\mathcal {R}$ and $\varphi$ is linear on $[0,\frac{1}{3}]$, on $[\frac{1}{3}, \frac{2}{3}]$ and on $[\frac{2}{3}, 1]$. Note that $\varphi ^{`\mathcal {R}}(0_\mathcal {R})\le _\mathcal {R}0_\mathcal {R}\le _\mathcal {R}\varphi ^{`\mathcal {R}}(1_\mathcal {R})$. Using ${\textbf {IVT}}$, find $\gamma$ in [0, 1] such that $\varphi ^{`\mathcal {R}}(\gamma ) =_\mathcal {R} 0_\mathcal {R}$. Either $\gamma > _\mathcal {R}\frac{1}{3}$ or $\gamma <_\mathcal {R} \frac{2}{3}$. If $\gamma > _\mathcal {R}\frac{1}{3}$, then $\lnot (\delta >_\mathcal {R} 0)$ and: $\forall p[2p\ne \mu n[\beta (n)\ne 0]]$, and, if $\gamma <_\mathcal {R} \frac{2}{3}$, then $\lnot (\delta <_\mathcal {R} 0)$ and $\forall p[2p+1\ne \mu n[\beta (n)\ne 0]]$. We thus see that $\forall \beta [\forall p[2p\ne \mu n[\beta (n)\ne 0]]\;\vee \;\forall p[2p+1\ne \mu n[\beta (n)\ne 0]]]$, i.e. ${\textbf {LLPO}}$. $\square$

Theorem 10.2

$\textsf{BIM}+ \varvec{\Pi }^0_1$-$\textbf{AC}_{\omega ,2}\vdash \textbf{LLPO}\rightarrow \textbf{IVT}$.

Proof

Let $\varphi$ in $\mathcal {R}^{[0,1]}$ and $\gamma$ in $[0,1]^2$ be given such that $\varphi ^{`\mathcal {R}}(\gamma ^{\upharpoonright 0} ) \le _\mathcal {R} 0_\mathcal {R}\le _\mathcal {R}\varphi ^{`\mathcal {R}}(\gamma ^{\upharpoonright 1})$. Define $\beta$ such that, for all n, $\beta (2n)\ne 0 \leftrightarrow \gamma ^{\upharpoonright 0}(n)<_\mathbb {S} \gamma ^{\upharpoonright 1}(n)$ and: $\beta (2n+1)\ne 0 \leftrightarrow \gamma ^{\upharpoonright 1}(n)<_\mathbb {S} \gamma ^{\upharpoonright 0}(n)$. By ${\textbf {LLPO}}$, either $\forall p[2p+1\ne \mu n[\beta (n)\ne 0]]$ and $\forall n[\gamma ^{\upharpoonright 0}(n) \le _\mathbb {S} \gamma ^{\upharpoonright 1}(n) ]$ and $\gamma ^{\upharpoonright 0}\le _\mathcal {R}\gamma ^{\upharpoonright 1}$, or $\forall p[2p\ne \mu n[\beta (n)\ne 0]]$ and $\forall n[\gamma ^{\upharpoonright 1}(n) \le _\mathbb {S} \gamma ^{\upharpoonright 0} (n) ]$ and $\gamma ^{\upharpoonright 1}\le _\mathcal {R}\gamma ^{\upharpoonright 0}$.

Assume $\gamma ^{\upharpoonright 0} \le _\mathcal {R} \gamma ^{\upharpoonright 1}$. Using ${\textbf {LLPO}}$ like we used it just now, conclude that, for all n, for all $m\le 2^n$, either $\varphi ^{`\mathcal {R}}(\frac{2^n-m}{2^n}\cdot _\mathcal {R}\gamma ^{\upharpoonright 0} +_\mathcal {R} \frac{m}{2^n}\cdot _\mathcal {R} \gamma ^{\upharpoonright 1}) \le _\mathcal {R} 0_\mathcal {R}$, or $0_\mathcal {R}\le _\mathcal {R} \varphi ^{`\mathcal {R}}(\frac{2^n-m}{2^n}\cdot _\mathcal {R}\gamma ^{\upharpoonright 0} +_\mathcal {R} \frac{m}{2^n}\cdot _\mathcal {R} \gamma ^{\upharpoonright 1})$. Using $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, find $\beta$ such that $\beta ^{\upharpoonright 0}(0)=0$ and $\beta ^{\upharpoonright 1}(0)\ne 0$ and for all n, for all $m\le 2^n$, if $\beta ^{\upharpoonright n}(m)=0$, then $\varphi ^{`\mathcal {R}}(\frac{2^n-m}{2^n}\cdot _\mathcal {R}\gamma ^{\upharpoonright 0} +_\mathcal {R} \frac{m}{2^n}\cdot _\mathcal {R} \gamma ^{\upharpoonright 1}) \le _\mathcal {R} 0_\mathcal {R}$, and, if $\beta ^{\upharpoonright n}(m)\ne 0$, then $0_\mathcal {R}\le _\mathcal {R}\varphi ^{`\mathcal {R}}(\frac{2^n-m}{2^n}\cdot _\mathcal {R}\gamma ^{\upharpoonright 0} +_\mathcal {R} \frac{m}{2^n}\cdot _\mathcal {R} \gamma ^{\upharpoonright 1})$. Now define $\delta$ such that $\delta (0) = 0$ and, for all n, if $\beta ^{\upharpoonright ( n+1)}\bigl (2\delta (n) +1\bigr )\ne 0$, then $\delta (n+1) =2 \delta (n)$, and, if $\beta ^{\upharpoonright (n+1)}\bigl (2\delta (n) +1\bigr )= 0$, then $\delta (n+1) =2 \delta (n)+1$. Note that, for all n, $\delta (n)<2^n$ and $\beta ^{\upharpoonright n}\bigl (\delta (n)\bigr ) = 0$ and $\beta ^{\upharpoonright n}\bigl (\delta (n)+1\bigr ) \ne 0$ and $\varphi ^{`\mathcal {R}}\bigl ((\frac{2^n-\delta (n)}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 0}+_\mathcal {R}(\frac{\delta (n)}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 1}\bigr )\le _\mathcal {R}0_\mathcal {R}\le _\mathcal {R}\varphi ^{`\mathcal {R}}\bigl ((\frac{2^n-\delta (n)-1}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 0}+_\mathcal {R}(\frac{\delta (n)+1}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 1}\bigr )$. Define $\varepsilon$ such that, for each n, $\varepsilon (n)=\bigl ((\frac{2^n-\delta (n)}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 0}+_\mathcal {R}(\frac{\delta (n)}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 1}, (\frac{2^n-\delta (n)-1}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 0}+_\mathcal {R}(\frac{\delta (n)+1}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 1}\bigr )$. Note that $\varepsilon \in [0,1]$. Assume $\varphi ^{`\mathcal {R}}(\varepsilon )>_\mathcal {R} 0_\mathcal {R}$. Find n, p such that $p\in E_\varphi$ and $\varepsilon (n)\sqsubseteq _\mathbb {S} p'$ and $(p'')'>_\mathbb {Q} 0_\mathbb {Q}$. Note that $\varepsilon '(n)\le _\mathbb {Q}\frac{2^n-\delta (n)}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 0}+_\mathcal {R}(\frac{\delta (n)}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 1}\le _\mathbb {Q}\varepsilon ''(n)$ and $\varphi ^{`\mathcal {R}}\bigl (( \frac{2^n-\delta (n)}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 0}+_\mathcal {R}(\frac{\delta (n)}{2^n})\cdot _\mathcal {R}\gamma ^{\upharpoonright 1}\bigr )\le _\mathbb {R} 0_\mathbb {R}$. Contradiction. Conclude that $\varphi ^{`\mathcal {R}}(\rho )\le _\mathcal {R} 0_\mathcal {R}$. For a similar reason, $0_\mathcal {R}\le _\mathcal {R}\varphi ^{`\mathcal {R}}(\rho )$ and, therefore, $\varphi ^{`\mathcal {R}}(\rho ) =_\mathcal {R} 0_\mathcal {R}$.

The case $\gamma ^{\upharpoonright 1} \ge _\mathcal {R}\gamma ^{\upharpoonright 0}$ is treated similarly.$\square$

Corollary 10.3

$\textsf{BIM}+\varvec{\Pi }^0_1$-$\textbf{AC}_{\omega ,2}\vdash \textbf{IVT} \leftrightarrow \textbf{LLPO} \leftrightarrow \textbf{WKL}$.

Proof

Use Theorems  4.3, 10.1 and 10.2. $\square$

10.2. A contraposition of the Intermediate Value Theorem, $\overleftarrow{{\textbf {IVT}}}$:

For each $\varphi$ in $\mathcal {R}^{[0,1]}$, if $\forall \gamma \in [0,1][\varphi ^{`\mathcal {R}}(\gamma )\; \#_\mathcal {R} \; 0_\mathcal {R}]$, then

either $\forall \gamma \in [0,1][0_\mathcal {R}<_\mathcal {R}\varphi ^{`\mathcal {R}}(\gamma )]$ or $\forall \gamma \in [0,1][ \varphi ^{`\mathcal {R}}(\gamma ) <_\mathcal {R} 0_\mathcal {R}].$

Theorem 10.4

$\textsf{BIM}\vdash \overleftarrow{\textbf{IVT}}$.

Proof

Assume $\varphi \in \mathcal {R}^{[0.1]}$ and $\forall \gamma \in [0,1][\varphi ^{`\mathcal {R}}(\gamma )\; \#_\mathcal {R} \;0_\mathcal {R}]$. Assume that $\varphi ^{`\mathcal {R}}(0_\mathcal {R}) <_\mathcal {R} 0_\mathcal {R}$. Suppose we find $\gamma$ in [0, 1] such that $0_\mathcal {R}<_\mathcal {R} \varphi ^{`\mathcal {R}}(\gamma )$. We will obtain a contradiction by the method of successive bisection. Find q in $\mathbb {Q}$ such that $0_\mathcal {R} <_\mathcal {R} \varphi ^{`\mathcal {R}}\bigl ((q)_\mathcal {R}\bigr )$. Define $\delta$ such that, for each n, $\delta (n) \in \mathbb {S}$ and $\delta (n)\sqsubseteq _\mathbb {S} (0,1)$, as follows, by induction. Define $\delta (0):= ( 0, q )$. Let n, s be given such that $\delta (n) = s$. Note that $\varphi ^{`\mathcal {R}}\bigl ((\frac{s'+_\mathbb {Q}s''}{2})_\mathcal {R}\bigr )\;\#\;0_\mathcal {R}$. Find (r, t) in $E_\varphi$ such that $r'<_\mathbb {Q} \frac{1}{2}(s'+_\mathbb {Q} s'') <_\mathbb {Q} r''$ and either $0_\mathbb {Q}<_\mathbb {Q} t'$ or $t''<_\mathbb {Q} 0_\mathbb {Q}$. If $0_\mathbb {Q}<_\mathbb {Q} t'$, define $\delta (n+1) = (s', \frac{s'+_\mathbb {Q}s''}{2})$, and, if $t''<_\mathbb {Q} 0_\mathbb {Q}$, define $\delta (n+1) = (\frac{s'+ s''}{2}, s'' )$. Note that, for each n, $\delta (n+1) \sqsubseteq _\mathbb {S} \delta (n)$, and $\varphi ^{`\mathcal {R}}\bigl ((\delta '(n))_\mathcal {R}\bigr )<_\mathcal {R} 0_\mathcal {R}<_\mathcal {R}\varphi ^{`\mathcal {R}}\bigl ((\delta ''(n))_\mathcal {R}\bigr )$. Note that $\delta \in [0,1]$ and $\varphi ^{`\mathcal {R}}(\delta )\;\#_\mathcal {R}\; 0_\mathcal {R}$. Determine (r, s) in $E_\varphi$ and n in $\omega$ such that $\delta (n) \sqsubseteq _\mathbb {S} r$ and either $s''<_\mathbb {Q} 0_\mathbb {Q}$ or $0_\mathbb {Q}<_\mathbb {Q} s'$, that is, either $\varphi ^{`\mathcal {R}}\bigl ((\delta ''(n))_\mathcal {R}\bigr ) <_\mathcal {R} 0_\mathcal {R}$ or $0_\mathcal {R} <_\mathcal {R}\varphi ^{`\mathcal {R}}\bigl ((\delta '(n))_\mathcal {R}\bigr )$. Contradiction. Conclude that $\lnot \bigl (\varphi ^{`\mathcal {R}}(\gamma ) >_\mathcal {R} 0_\mathcal {R}\bigr )$. As $\varphi ^{`\mathcal {R}}(\gamma ) \;\#_\mathcal {R}\; 0_\mathcal {R}$, conclude that $\varphi ^{`\mathcal {R}}(\gamma ) <_\mathcal {R} 0_\mathcal {R}$. We thus see that, if $\varphi ^{`\mathcal {R}}(0_\mathcal {R}) < _\mathcal {R}0_\mathcal {R}$, then $\forall \gamma \in [0,1][\varphi ^{`\mathcal {R}}(\gamma ) <_\mathcal {R} 0_\mathcal {R}]$. One may prove also that, if $\varphi ^{`\mathcal {R}}(0_\mathcal {R}) >_\mathcal {R} 0_\mathcal {R}$, then $\forall \gamma \in [0,1][\varphi ^{`\mathcal {R}}(\gamma ) >_\mathcal {R} 0_\mathcal {R}]$. Conclude that either $\forall \gamma \in [0,1][0_\mathcal {R}<_\mathcal {R}\varphi ^{`\mathcal {R}}(\gamma )]$ or $\forall \gamma \in [0,1][ \varphi ^{`\mathcal {R}}(\gamma ) <_\mathcal {R} 0_\mathcal {R}].$ $\square$

10.1 ${\textbf {FT}}$ is unprovable in $\textsf{BIM}+ {\textbf {IVT}}$

As we observed in Sect. 4.8, $\textsf{BIM}+{\textbf {CT}}+X\vee \lnot X$ is consistent. According to Theorem 10.4, $\textsf{BIM}\vdash \overleftarrow{{\textbf {IVT}}}$. Conclude that $\textsf{BIM} + X\vee \lnot X\vdash {\textbf {IVT}}$. Assume that $\textsf{BIM}\vdash {\textbf {IVT}} \rightarrow {\textbf {FT}}$. Then $\textsf{BIM} + X\vee \lnot X\vdash {\textbf {FT}}$. As we know from Theorem 2.5, $\textsf{BIM} + {\textbf {CT}}\vdash \lnot !{\textbf {FT}}$, and, therefore, $\textsf{BIM} + {\textbf {CT}}\vdash \lnot {\textbf {FT}}$. Conclude that $\textsf{BIM}+{\textbf {IVT}}\nvdash {\textbf {FT}}$, and also, in view of Theorem 2.6, $\textsf{BIM}+{\textbf {IVT}}\nvdash {\textbf {WKL}}$. Note that, in view of Corollary  10.3, this gives another proof of $\textsf{BIM}\nvdash \varvec{\Pi }_1^0$-${\textbf {AC}}_{\omega , 2}$, a fact established in Sect. 4.8. One may even conclude that $\textsf{BIM}+{\textbf {IVT}}\nvdash \varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$.

One may ask if $\textsf{BIM} +{\textbf {LLPO}}\vdash {\textbf {IVT}}$, i.e. if the proof of Theorem 10.2 can be given without recourse to $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,2}$, but we do not know the answer to this question.

10.4. The Uniform Intermediate Value Theorem, ${\textbf {UIVT}}$:

For each $\varphi$ such that $\forall n[\varphi ^n\in \mathcal {R}^{[0,1]}]$,

if $\forall n \exists \gamma \in [0,1]^2[\ (\varphi ^n)^{`\mathcal {R}}(\gamma ^0 ) \le _\mathcal {R} 0_\mathcal {R}\le _\mathcal {R} (\varphi ^n)^{`\mathcal {R}} (\gamma ^1)]$,

then $\exists \gamma \in [0,1]^\omega \forall n[(\varphi ^n)^{`\mathcal {R}}(\gamma ^n) =_\mathcal {R} 0_\mathcal {R}].$

In [27, Exercise IV.2.12, page 137], the reader is asked to prove that, in the classical system $\mathsf {RCA_0}$, ${\textbf {UIVT}}$ is an equivalent of ${\textbf {WKL}}$. As $\mathsf {RCA_0}\vdash {\textbf {IVT}}$, $\mathsf {RCA_0}$ proves the equivalence of ${\textbf {UIVT}}$ and the next principle.

10.5. ${\textbf {Uzero}}$:

For all $\varphi$ such that $\forall n[\varphi ^{\upharpoonright n} \in \mathcal {R}^{[0,1]}]$,

if $\forall n\exists \gamma \in [0,1][\ (\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) =_\mathcal {R} 0_\mathcal {R}]$, then $\exists \gamma \in [0,1]^\omega \forall n[(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ^{\upharpoonright n}) =_\mathcal {R} 0_\mathcal {R}].$

We want to study ${\textbf {Uzero}}$ in $\textsf{BIM}$. We need the following Lemma.

Lemma 10.5

$\textsf{BIM}$ proves:

• (i) $\exists \psi :\omega ^\omega \rightarrow \omega ^\omega \forall \varphi \in \mathcal {R}^{[0,1]}[\mathcal {H}_{\psi |\varphi }=\{\gamma \in [0,1]\mid \varphi ^{`\mathcal {R}}(\gamma ) \;\#_\mathcal {R}\;0_\mathcal {R}\}]$, and

• (ii) $\exists \tau :\omega ^\omega \rightarrow \omega ^\omega \forall \alpha [\tau |\alpha \in \mathcal {R}^{[0,1]}\;\wedge \;\mathcal {H}_\alpha =\{\gamma \in [0,1]\mid (\tau |\alpha )^{`\mathcal {R}}(\gamma )\;\#_\mathcal {R}\;0_\mathcal {R}\}]$.

Proof

(i) Define $\psi :\omega ^\omega \rightarrow \omega ^\omega$ such that, for each $\varphi$, for each s,

$$\begin{aligned} (\psi |\varphi )(s) \ne 0\leftrightarrow \exists p\in E_{{\overline{\varphi }} s}[s\sqsubset _\mathbb {S}p'\;\wedge \;\bigl (0_\mathbb {Q}<_\mathbb {Q} (p'')'\;\vee \; (p'')''<_\mathbb {Q} 0_\mathbb {Q}\bigr )]. \end{aligned}$$

Note that $\forall \varphi \in \mathcal {R}^{[0,1]}\forall \gamma \in [0,1][\varphi ^{`\mathcal {R}}(\gamma ) \;\#_\mathcal {R}\; 0_\mathcal {R}\leftrightarrow \gamma \in \mathcal {H}_{\psi |\varphi }]$.

(ii) Define $\rho$ such that, for each s in $\mathbb {S}$, $\rho ^s\in \mathcal {R}^{[0,1]}$ and, if not $(-1)_\mathbb {Q}\le _\mathbb {Q}s'\le _\mathbb {Q} s''\le _\mathbb {Q} (2)_\mathbb {Q}$, then, for all $\gamma$ in [0, 1], $(\rho ^s)^{`\mathcal {R}}(\gamma ) =_\mathcal {R} 0_\mathcal {R}$, and, if $(-1)_\mathbb {Q}\le _\mathbb {Q}s'\le _\mathbb {Q} s''\le _\mathbb {Q} (2)_\mathbb {Q}$, then, for all $\gamma$ in [0, 1],

• (1) if $\gamma \le _\mathcal {R} (s')_\mathcal {R}$ or $(s'')_\mathcal {R} \le _\mathcal {R} \gamma$, then $\rho ^s(\gamma ) =_\mathcal {R} 0_\mathcal {R}$, and,

• (2) if $(s')_\mathcal {R}\le _\mathcal {R} \gamma \le _\mathcal {R} (s'')_\mathcal {R}$, then $\rho ^s(\gamma ) =_\mathcal {R} \inf (\gamma -_\mathcal {R} (s')_\mathcal {R},(s'')_\mathcal {R}-_\mathcal {R} \gamma )$.

(Note that $\rho ^s$ codes the restriction to [0, 1] of the ‘tent’ function from $\mathcal {R}$ tot $\mathcal {R}$ that is zero outside of $[s', s'']$ and linear on both $[s', \frac{s'+s''}{2}]$ and $[\frac{s'+s''}{2}, s'']$ and that takes the value $0_\mathcal {R}$ at $s'$ and the value $\frac{s''-s'}{2}$ at $\frac{s'+s''}{2}$ and the value $0_\mathcal {R}$ at $s''$. Note that, for all s in $\mathbb {S}$, for all $\gamma$ in [0, 1], $(\rho ^s)^{`\mathcal {R}}(\gamma )\le _\mathcal {R} (\frac{3}{2})_\mathcal {R}$. )

Define $\tau :\omega ^\omega \rightarrow \omega ^\omega$ such that, for all $\alpha$, $\tau |\alpha \in \mathcal {R}^{[0,1]}$ and, for each $\gamma$ in [0, 1], $(\tau |\alpha )^{`\mathcal {R}}(\gamma ) =_\mathcal {R} \sum _{s, \alpha (s)\ne 0} (\frac{1}{2^s})_\mathcal {R}\cdot _\mathcal {R}(\rho ^s)^{`\mathcal {R}}(\gamma )$. Note that $\forall \gamma \in [0,1][ \gamma \in \mathcal {H}_{\alpha }\leftrightarrow (\tau |\alpha )^{`\mathcal {R}}(\gamma ) \;\#_\mathcal {R} \;0_\mathcal {R}]$, and that $\forall \gamma \in [0,1][\gamma \notin \mathcal {H}_{\alpha } \leftrightarrow (\tau |\alpha )^{`\mathcal {R}}(\gamma ) =_\mathcal {R} 0_\mathcal {R}]$.$\square$

Theorem 10.6

$\textsf{BIM}\vdash \varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,[0,1]}\leftrightarrow \textbf{Uzero}$.

Proof

First, assume $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,[0,1]}$. Let $\varphi$ be given such that $\forall n[\varphi ^{\upharpoonright n} \in \mathcal {R}^{[0,1]} \;\wedge \; \exists \gamma \in [0,1][(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) =_\mathcal {R} 0_\mathcal {R}]]$. Using Lemma 10.5(i), find $\alpha$ such that, $\forall n[\mathcal {H}_{\alpha ^{\upharpoonright n}}=\{\gamma \in [0,1]\mid (\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma )\;\#_\mathcal {R}\; 0_\mathcal {R}\}]$. Conclude that $\forall n \exists \gamma \in [0,1] [\gamma \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$. Using $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,[0,1]}$ conclude that $\exists \gamma \in [0,1]^\omega \forall n[\gamma ^{\upharpoonright n} \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$ and

$\exists \gamma \in [0,1]^\omega \forall n[(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ^{\upharpoonright n}) =_\mathcal {R} 0_\mathcal {R}]$. We thus see ${\textbf {Uzero}}$.

Secondly, assume ${\textbf {Uzero}}$. Let $\alpha$ be given such that $\forall n\exists \gamma \in [0,1][\gamma \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$. Using Lemma 10.5(ii), find $\varphi$ such that $\forall n[\varphi ^{\upharpoonright n}\in \mathcal {R}^{[0,1]}]$ and $\forall n[\mathcal {H}_{\alpha ^{\upharpoonright n}}=\{\gamma \in [0,1]\mid (\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) \;\#_\mathcal {R} \;0_\mathcal {R}\}]$. Conclude that $\forall n\exists \gamma \in [0,1][(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) =_\mathcal {R} 0_\mathcal {R}]$. Using ${\textbf {Uzero}}$ conclude that $\exists \gamma \in [0,1]^\omega \forall n[(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ^{\upharpoonright n}) =_\mathcal {R} 0_\mathcal {R}]$, and $\exists \gamma \in [0,1]^\omega \forall n [\gamma ^{\upharpoonright n} \notin \mathcal {H}_{\alpha ^{\upharpoonright n}}]$. We thus see $\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,[0,1]}$. $\square$

10.6. A uniform contrapositive Intermediate Value Theorem $\overleftarrow{{\textbf {UIVT}}}:$

For each $\varphi$ such that $\forall n[\varphi ^{\upharpoonright n}\in \mathcal {R}^{[0,1]}]$, $\textit{if}\; \forall \gamma \in [0,1]^{\omega } \exists n[(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ^{\upharpoonright n})\; \#_\mathcal {R} \; 0_\mathcal {R}],$

$\textit{then}\; \exists n[ \forall \gamma \in [0,1][(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) > _\mathcal {R}0_\mathcal {R}]\;\vee \; \forall \gamma \in [0,1][(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) <_\mathcal {R} 0_\mathcal {R}]].$

As $\textsf{BIM}\vdash \overleftarrow{{\textbf {IVT}}}$, $\textsf{BIM}$ proves the equivalence of $\overleftarrow{{\textbf {UIVT}}}$ and the next statement.

10.7. $\overleftarrow{{\textbf {Uzero}}}$: For all $\varphi$ such that $\forall n[\varphi ^{\upharpoonright n}\in \mathcal {R}^{[0,1]}]$,

$\textit{if}\;\forall \gamma \in [0,1]^{\omega } \exists n[(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ^{\upharpoonright n})\; \#_\mathcal {R} \; 0_\mathcal {R}], \;\textit{then}\; \exists n \forall \gamma \in [0,1][(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) \;\# _\mathcal {R}\;0_\mathcal {R}].$

We define a strong negation of this statement. This strong negation itself contains a negation sign, a possibility mentioned in Sect. 1.4.

10.8. $\lnot !\overleftarrow{{\textbf {Uzero}}}$:

There exists $\varphi$ such that $\forall n[\varphi ^{\upharpoonright n}\in \mathcal {R}^{[0,1]}]$ and

$\forall \gamma \in [0,1]^{\omega } \exists n[(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ^{\upharpoonright n})\; \#_\mathcal {R} \; 0_\mathcal {R}]$ and $\lnot \exists n \forall \gamma \in [0,1][(\varphi ^n)^{`\mathcal {R}}(\gamma ) \;\# _\mathcal {R}\;0_\mathcal {R}]$.

Lemma 10.7

One may prove in $\textsf{BIM}$:

• (i) $\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}}\rightarrow \overleftarrow{\textbf{Uzero}}$ and $\lnot !\overleftarrow{\textbf{Uzero}} \rightarrow \lnot !\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}}$.

• (ii) $\overleftarrow{\textbf{Uzero}} \rightarrow \varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}}$ and $\lnot !\varvec{\Sigma }^0_1$-$\overleftarrow{\textbf{AC}_{\omega ,[0,1]}}\rightarrow \lnot !\overleftarrow{\textbf{Uzero}}$

Proof

(i) The two promised conclusions follow if one may prove in $\textsf{BIM}$ that, for every $\varphi$, if $\forall n[\varphi ^{\upharpoonright n} \in \mathcal {R}^{[0,1]}]$, then there exists $\beta$ such that

$$\begin{aligned}{} & {} \forall \gamma \in [0,1]^\omega \exists n[(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ^{\upharpoonright n}) \;\#_\mathcal {R}\; 0_\mathcal {R}] \rightarrow \forall \gamma \in [0,1]^\omega \exists n [\gamma ^{\upharpoonright n} \in \mathcal {H}_{\beta ^{\upharpoonright n}}]\;\text {and}\\{} & {} \exists n [ [0,1] \subseteq \mathcal {H}_{\beta ^{\upharpoonright n}}] \rightarrow \exists n\forall \gamma \in [0,1] [(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) \;\#_\mathcal {R}\; 0_\mathcal {R}]. \end{aligned}$$

Using Lemma 10.5(i), one finds $\beta$ such that

$\forall n [\mathcal {H}_{\beta ^{\upharpoonright n}}=\{\gamma \in [0,1]\mid (\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) \;\#_\mathcal {R} \; 0_\mathcal {R}\}]$. The two statements now are obvious.

(ii) The two promised conclusions follow if one may prove in $\textsf{BIM}$ that, for each $\alpha$, there exists $\varphi$ such that $\forall n[\varphi ^{\upharpoonright n} \in \mathcal {R}^{[0,1]}]$ and

$$\begin{aligned}{} & {} \forall \gamma \in [0,1]^\omega \exists n [\gamma ^{\upharpoonright n} \in \mathcal {H}_{\alpha ^{\upharpoonright n}}] \rightarrow \forall \gamma \in [0,1]^\omega \exists n[(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ^{\upharpoonright n}) \;\#_\mathcal {R}\; 0_\mathcal {R}]\;\text {and}\\{} & {} \exists n\forall \gamma \in [0,1] [(\varphi ^{\upharpoonright n})^{`\mathcal {R}}(\gamma ) \;\#_\mathcal {R}\; 0_\mathcal {R}]) \rightarrow \exists n [ [0,1] \subseteq \mathcal {H}_{\alpha ^{\upharpoonright n}}]. \end{aligned}$$

Let $\alpha$ be given. Using Lemma 10.5(ii), find $\varphi$ such that $\forall n[\varphi ^n \in \mathcal {R}^{[0,1]}\;\wedge \; \mathcal {H}_{\alpha ^n} =\{\gamma \in [0,1]\mid (\varphi ^n)^{`\mathcal {R}}(\gamma ) \;\#_\mathcal {R}\; 0_\mathcal {R}\}]$. The two statements now are obvious. $\square$

Theorem 10.8

$\textsf{BIM}$ proves $\overleftarrow{\textbf{Uzero}}\leftrightarrow \overleftarrow{\textbf{UIVT}}\leftrightarrow \textbf{FT}$ and

$\lnot !\overleftarrow{\textbf{Uzero}}\leftrightarrow \lnot !\overleftarrow{\textbf{UIVT}} \leftrightarrow \lnot !\textbf{FT}$.

Proof

Use Lemma 10.7 and Theorem 7.2. $\square$

11 The compactness of classical propositional logic

In this Section, we prove that ${\textbf {FT}}$ is equivalent to a contraposition of a restricted version of the compactness theorem for classical propositional logic. We also prove the corresponding result for $\lnot !{\textbf {FT}}$.

We introduce the symbols $\lnot$, $\bigwedge$ and $\bigvee$ as natural numbers: $\lnot := 1$, $\bigwedge := 2$ and $\bigvee := 3$. We define (the characteristic function of) $\textit{Form}\subseteq \omega$, as follows, by recursion. For each n, $n\in \textit{Form}$ if and only if

$$\begin{aligned}{} & {} n'=0 \;\vee \; (n'=\lnot \;\wedge \; n''\in \textit{Form}) \;\vee \;\\{} & {} \bigl ((n'=\bigwedge \;\vee \;n'=\bigvee ) \;\wedge \;\forall i<\textit{length}(n'')[n''(i)\in \textit{Form}]\bigr ). \end{aligned}$$

We define $\top :=(\bigwedge , \langle \;\rangle )$ and $\perp :=(\bigvee , \langle \;\rangle )$.

Assume $\gamma \in 2^\omega$. We define ${\tilde{\gamma }}$ in $2^\omega$ such that, for every n,

• (i) if $n\notin \textit{Form}$, then ${\tilde{\gamma }} (n) =0$, and,

• (ii) if $n\in \textit{Form}$ and $n'=0$, then ${\tilde{\gamma }} (n) = \gamma (n'')$, and,

• (ii) if $n\in \textit{Form}$ and $n'=\lnot$, then ${\tilde{\gamma }}(n) = 1 - {\tilde{\gamma }} (n'')$, and,

• (iii) if $n\in \textit{Form}$ and $n'=\bigwedge$, then ${\tilde{\gamma }} (n)= \min \{{\tilde{\gamma }}\bigl (n''(i)\bigr )|i< \textit{length}(n'')\}$, and

• (iii) if $n\in \textit{Form}$ and $n'=\bigvee$, then ${\tilde{\gamma }} (n)= \max \{{\tilde{\gamma }}\bigl (n''(i)\bigr )|i< \textit{length}(n'')\}$.

Note that $0 = \langle \;\rangle$. We define $\min (\emptyset )=1$ and $\max (\emptyset )=0$. Note that ${\tilde{\gamma }}(\top ) = {\tilde{\gamma }}\bigl ((\bigwedge , 0)\bigr ) = 1$ and ${\tilde{\gamma }}(\perp ) ={\tilde{\gamma }}\bigl ((\bigvee , 0)\bigr ) = 0$. For all m, n in Form, we define: $m\equiv n$ if and only if $\forall \gamma \in 2^\omega [{\tilde{\gamma }}( m) ={\tilde{\gamma }} (n)]$.

Assume $c \in 2^{<\omega }$. We define ${\tilde{c}}$ in $2^{<\omega }$ such that $\textit{length}(c) = \textit{length}({\tilde{c}})$, as follows. First, define $\gamma =c*\underline{0}$. Then define, for all $m<length(c)$, ${\tilde{c}} (m):={\tilde{\gamma }} (m)$.

$X\subseteq \omega$ is realizable, $\textit{Real}(X)$, if and only if $\exists \gamma \in 2^\omega \forall n \in X[{\tilde{\gamma }}(n) =1]$, and positively unrealizable, $\textit{Unreal}(X)$, if and only if $\forall \gamma \in 2^\omega \exists n \in X[{\tilde{\gamma }} (n) = 0]$.

We define a mapping $\textit{Fm}$ from $2^{<\omega }$ to $\textit{Form}$, as follows. Assume $a \in 2^{<\omega }$. Find s such that $\textit{length}(s) = \textit{length}(a)$, and, for all $i < \textit{length}(a)$, if $a(i) = 0$, then $s(i) = (\lnot , (0,i))$, and, if $a(i) = 1$, then $s(i) = (0,i)$. Define $Fm(a)= (\bigwedge , s)$.

Lemma 11.1

• (i) $\forall a \in 2^{<\omega }\forall \gamma \in 2^\omega [{\tilde{\gamma }}\bigl (\textit{Fm}(a)\bigr ) = 1\leftrightarrow a \sqsubset \gamma ]$.

• (ii) There exists $\beta$ such that $\forall m \in Form \forall p[\beta \bigl (m,p)\bigr )>p\;\wedge \; \beta \bigl ((m,p)\bigr ) \in Form\;\wedge \; \beta \bigl ((m,p)\bigr )\equiv m]$.

• (iii) ²³ For all $\alpha$ in $2^\omega$, there exists $\delta$ in $[\omega ]^\omega$ such that $\forall m[\delta (m)\in \textit{Form}\;\wedge \;\forall \gamma \in 2^\omega [{\tilde{\gamma }}\bigl (\delta (m)\bigr )=1 \leftrightarrow \forall n\le m[\alpha ({\overline{\gamma }} n)=0]]]$.

Proof

(i) We prove, by induction, that, for each n, $\forall a \in 2^{<\omega }[length(a)=n \rightarrow \forall \gamma \in 2^\omega [{\tilde{\gamma }}\bigl (\textit{Fm}(a)\bigr ) = 1\leftrightarrow a \sqsubset \gamma ]]$. Note that $Fm(\langle \;\rangle )=\top$ and $\forall \gamma \in 2^\omega [{\tilde{\gamma }}(\top )=1]$ and $\forall \gamma \in 2^\omega [\langle \;\rangle \sqsubset \gamma ]$. Now let a, n be given such that $a\in 2^{<\omega }$ and $length(a)=n$ and $\forall \gamma \in 2^\omega [{\tilde{\gamma }}\bigl (\textit{Fm}(a)\bigr ) = 1\leftrightarrow a \sqsubset \gamma ]$. Note that for each $\gamma$ in $2^\omega$, for each $i<2$, ${\tilde{\gamma }}\bigl (Fm(a*\langle i\rangle )\bigr )=1\leftrightarrow \bigl ({\tilde{\gamma }}\bigl (Fm(a)\bigr ) = 1 \;\wedge \; \gamma (n)=i \bigr )\leftrightarrow a*\langle i \rangle \sqsubset \gamma$.

(ii) The proof is an exercise in calculating codes of formulas. Given m in Form and p, one might first find $q:=\max (m,p)$ and then s in $\omega ^{q+1}$ such that $s(0)=m$ and $\forall j<q[s(j)=\top ]$ and then define $\beta \bigl ((m,p)\bigr ):=(\bigwedge , s)$.

(iii) Let $\alpha$ be given. We define the promised $\delta$ as follows, by induction. If $\alpha (\langle \;\rangle )=0$, define $\delta (0):=\top$, and, if $\alpha (\langle \;\rangle )\ne 0$, define $\delta (0):=\perp$. Note that $\delta (0)$ satisfies the requirements. Let m be given such that $\delta (m)$ has been defined and ${\overline{\delta }}(m+1)\in [\omega ]^{m+1}$. Find t such that $\{t(i)\mid i< \textit{length}(t)\}=\{a\in 2^{<\omega }\mid length(a)= m+1\;\wedge \;\forall n\le m+1[\alpha ({\overline{a}} n) = 0]\}$. Then find s such that $\textit{length}(s) = \textit{length}(t)$ and $\forall i< \textit{length}(s)[s(i) = \textit{Fm}\bigl (t(i)\bigr )]$. Note that, for each $\gamma$ in $2^\omega$, ${\tilde{\gamma }}\bigl ((\bigvee , s)\bigr )=1\leftrightarrow$ $\exists a \in 2^{<\omega }[length(a)=m+1 \;\wedge \;\forall n \le m[\alpha ({\overline{a}} n) = 0]\;\wedge \;{\tilde{\gamma }}\bigl (Fm(a)\bigr )=1]\leftrightarrow$ $\exists a \in 2^{<\omega }[length(a)=m+1\;\wedge \;\forall n \le m[\alpha ({\overline{a}} n) = 0]\;\wedge \;a \sqsubset \gamma ]\leftrightarrow \forall n\le m[\alpha ({\overline{\gamma }} n) =0]$. Define $\delta (m+1) =\beta \bigl ( (\bigvee , s), \delta (m)\bigr )$, where $\beta$ is the function we found in (ii). Note that $\delta (m+1)$ satisfies the requirements. $\square$

Lemma 11.2

The following statements are provable in $\textsf{BIM}$.

• (i) $\textbf{FT} \rightarrow \forall \alpha [\textit{Unreal}(E_\alpha ) \rightarrow \exists n[\textit{Unreal}(E_{{\overline{\alpha }} n})]]$.

• (ii) $\exists \alpha [\textit{Unreal}(E_\alpha ) \;\wedge \; \forall n[\textit{Real}(E_{{\overline{\alpha }} n})]]\rightarrow \lnot !\textbf{FT}$.

• (iii) $\textbf{WKL}\rightarrow \forall \alpha [\forall n [Real(E_{{\overline{\alpha }} n})]\rightarrow Real(E_\alpha )]]$.

• (iv) $\forall \alpha [\textit{Unreal}(D_\alpha ) \rightarrow \exists n[\textit{Unreal}(D_{{\overline{\alpha }} n})]]\rightarrow \textbf{FT}$.

• (v) $\lnot !\textbf{FT} \rightarrow \exists \alpha [\textit{Unreal}(D_\alpha ) \;\wedge \; \forall n[\textit{Real}(D_{{\overline{\alpha }} n})]]$.

• (vi) $\forall \alpha [\forall n[Real(D_{{\overline{\alpha }} n})]\rightarrow Real(D_\alpha )]]\rightarrow \textbf{WKL}$.

Proof

(i), (ii) and (iii). We argue in $\textsf{BIM}$.

Let $\alpha$ be given. Define $\beta$ such that, for all m, for all c in $2^{<\omega }$ such that $length(c)=m$, $\beta (c) \ne 0\leftrightarrow \exists n < m[n\in E_{{\overline{\alpha }} m}\;\wedge \;{\tilde{c}}(n) = 0]]$. We shall prove that

$$\begin{aligned}{} & {} \textit{Unreal}(E_\alpha ) \rightarrow \textit{Bar}_{2^\omega }(D_\beta )\;\text {and}\\{} & {} \exists m[ Bar_{2^\omega }(D_{{\overline{\beta }} m})]\rightarrow \exists n[\textit{Unreal}(E_{{\overline{\alpha }} n})]. \end{aligned}$$

Assume that $\textit{Unreal}(E_\alpha )$. Let $\gamma$ in $2^\omega$ be given. Find n, p such that $n \in E_{{\overline{\alpha }} p}$ and ${\tilde{\gamma }} (n) = 0$. Define $m:=\max \{n, p\}+1$ and note that $\beta ({\overline{\gamma }} m) \ne 0$. Conclude that $\forall \gamma \in 2^\omega \exists m[\beta ({\overline{\gamma }} m)\ne 0]$ and $\textit{Bar}_{2^\omega }(D_\beta )$.

Let m be given such that $Bar_{2^\omega }(D_{{\overline{\beta }} m})$. For all c in $2^{<\omega }$ such that $length(c)=m$, $\exists n\le m[\beta ({\overline{c}} n) \ne 0]$ and $\exists n < m[n \in E_{{\overline{\alpha }} m}\;\wedge \;{\tilde{c}}(n) = 0]$. Conclude that $\textit{Unreal}(E_{{\overline{\alpha }} m})$ and $\exists n[\textit{Unreal}(E_{{\overline{\alpha }} n})]$.

Note that, if $Unreal(E_\alpha )$, then $Bar_{2^\omega }(D_\beta )$, and by ${\textbf {FT}}$, there exist m such that $Bar_{2^\omega }(D_{{\overline{\beta }} m})$ and n such that $Unreal(E_{{\overline{\alpha }} n})$. This establishes (i).

Note that, if $Unreal(E_\alpha )\;\wedge \;\forall n[Real(E_{{\overline{\alpha }} n}]$, then $Bar_{2^\omega }(D_\beta )$ and

$\forall m[\lnot Bar_{2^\omega }(D_{{\overline{\beta }} m})]$, i.e. $\lnot {\textbf {! FT}}$. This establishes (ii).

Note that, if $\forall n[Real(E_{{\overline{\alpha }} n})]$, then $\forall m[\lnot Bar(D_{{\overline{\beta }} m})]$, and, by ${\textbf {WKL}}$, there exists $\gamma$ such that $\forall n[\beta ({\overline{\gamma }} n)=0 ]$. Conclude that $\forall m \forall n<m[n\in E_{{\overline{\alpha }} m}\rightarrow {\tilde{\gamma }}(n) =1]$, i.e. $\gamma$ realizes $E_\alpha$ and $Real(E_\alpha )$. This establishes (iii).

(iv), (v) and (vi). We argue in $\textsf{BIM}$.

Let $\alpha$ be given. Using Lemma 11.1(iii), find $\delta$ in $[\omega ]^\omega$ such that $\forall m[\delta (m)\in \textit{Form}\;\wedge \;\forall \gamma \in 2^\omega [{\tilde{\gamma }}\bigl (\delta (m)\bigr )=1] \leftrightarrow \forall n\le m[\alpha ({\overline{\gamma }} n)=0]]]$. Note that $\delta$ is strictly increasing and $\forall m[\exists n[m=\delta (n)]\leftrightarrow \exists n\le m[m=\delta (n)]]$. Define $\beta$ such that $\forall m[\beta (m) \ne 0\leftrightarrow \exists n[m = \delta (n)]]$. We shall prove that

$$\begin{aligned}{} & {} \textit{Bar}_{2^\omega }( D_\alpha )\rightarrow \textit{Unreal}(D_{ \beta })\;\text {and}\\{} & {} \exists n[\textit{Unreal}(D_{{\overline{\beta }} n})] \rightarrow \exists m [Bar_{2^\omega }(D_{{\overline{\alpha }} m})]. \end{aligned}$$

Assume that $\textit{Bar}_{2^\omega }(D_\alpha )$. Given any $\gamma \in 2^\omega$, find m such that $\alpha ({\overline{\gamma }} m) \ne 0$ and, therefore, ${\tilde{\gamma }}\bigl (\delta (m)\bigr )=0$ and: $\exists n \in D_\beta [{\tilde{\gamma }}(n) \ne 1]$. Conclude that $\textit{Unreal}(D_\beta )$.

Let n be given such that $\textit{Unreal}(D_{{\overline{\beta }} n})$. Let $m_0$ be the largest m such that $\delta (m)< n$. Note: $\lnot \exists \gamma \in 2^\omega [{\tilde{\gamma }}\bigl (\delta (m_0)\bigr )=1]$, and, therefore, $\forall a \in 2^{<\omega }[length(a)= m_0+1\rightarrow \exists n \le m_0[ \alpha ({\overline{a}} n) \ne 0]]$. Find k such that $\forall a\in 2^{<\omega }[length(a)= m_0+1\rightarrow a\le k]$ and conclude that $Bar_{2^\omega }(D_{{\overline{\alpha }} k})$ and $\exists m[Bar_{2^\omega }(D_{{\overline{\alpha }} m})]$.

Note that, if $Bar_{2^\omega }(D_\alpha )$ and $Unreal(D_\beta )\rightarrow \exists n[Unreal(D_{{\overline{\beta }} n})]$,

then $\exists m[Bar_2^\omega (D_{{\overline{\alpha }} m})]$. This establishes (iv).

Note that, if $Bar_{2^\omega }(D_\alpha )$ and $\lnot \exists n[Bar_{2^\omega }(D_{{\overline{\alpha }} n}]$, then $Unreal(D_\beta$

and $\forall n[Real(D_{{\overline{\beta }} n})]$. This establishes (v).

Note that, if $\forall n[\lnot Bar_2^\omega (D_{{\overline{\alpha }} n}]$ and $\forall n[Real(D_{{\overline{\beta }} n})]\rightarrow Real(D_\beta )$, then there exists $\gamma$ in $2^\omega$ realizing $D_\beta$, so $\forall m[{\tilde{\gamma }}\bigl (\delta (m)\bigr )=1]$ and $\forall m\forall n\le m[\alpha ({\overline{\gamma }} n)=0]$, i.e. $\forall n[\alpha ({\overline{\gamma }} n)=0]$. This establishes (vi). $\square$

Theorem 11.3

• (i) $\textsf{BIM}\vdash \textbf{FT} \leftrightarrow \forall \alpha [\textit{Unreal}(E_\alpha ) \rightarrow \exists n[\textit{Unreal}(E_{{\overline{\alpha }} n})]] \leftrightarrow \forall \alpha [\textit{Unreal}(D_\alpha ) \rightarrow \exists n[\textit{Unreal}(D_{{\overline{\alpha }} n})]]$.

• (ii) $\textsf{BIM}\vdash \lnot !\textbf{FT} \leftrightarrow \exists \alpha [\textit{Unreal}(E_\alpha ) \;\wedge \; \forall n[\textit{Real}(E_{{\overline{\alpha }} n})]] \leftrightarrow \exists \alpha [\textit{Unreal}(D_\alpha ) \;\wedge \; \forall n[\textit{Real}(D_{{\overline{\alpha }} n})]]$.

• (iii) $\textsf{BIM}\vdash \textbf{WKL} \leftrightarrow \forall \alpha [ \forall n[\textit{Real}(E_{{\overline{\alpha }} n})]\rightarrow Real(E_\alpha )] \leftrightarrow \forall \alpha [ \forall n[\textit{Real}(D_{{\overline{\alpha }} n})]\rightarrow Real(D_\alpha )]$.

Proof

Use Lemma 11.2 and the fact that $\forall \alpha \exists \beta [D_\alpha = E_\beta ]$. $\square$

Theorem 11.3(i) also is a consequence of [20, Theorem 6.5].

V.N. Krivtsov has shown, among other things, that ${\textbf {FT}}$ is an equivalent of an intuitionistic (generalized) completeness theorem for intuitionistic first-order predicate logic, see [18].

12 Other ‘Fan Theorems’?

In this Section, we indicate what, on our opinion, should be the subject of the next chapter in intuitionistic reverse mathematics. The Fan Theorem may be seen as a replacement, for the intuitionistic mathematician, of that enviable tool of the classical mathematician: (Weak) König’s Lemma. We hope to make clear that the greater subtlety of the language of the intuitionistic mathematician allows for many other possible replacements.

12.1 Notions of finiteness

Let $\alpha$ be given. We consider the set $D_\alpha :=\{n\mid \alpha (n)\ne 0\}$, the subset of $\omega$ decided by $\alpha$. We define the following.

$D_\alpha$ is finite if and only if $\exists n\forall m\ge n[\alpha (m)=0]$.

$D_\alpha$ is bounded-in-number if and only if $\exists n \forall t \in [\omega ]^{n+1} \exists i<n+1[\alpha \circ t(i)=0]$.

$D_\alpha$ is almost-finite if and only if $\forall \zeta \in [\omega ]^\omega \exists n[\alpha \circ \zeta (n)=0]$.

$D_\alpha$ is not-not-finite if and only if $\lnot \lnot \exists n\forall m\ge n[\alpha (m)=0]$.

$D_\alpha$ is infinite if and only if if $\forall n \exists m\ge n[\alpha (m)\ne 0]$.

$D_\alpha$ is not-infinite²⁴ if and only if $\lnot \forall n \exists m\ge n[\alpha (m)\ne 0]$.

Note that $D_\alpha$ is infinite if and only if $\exists \zeta \in [\omega ]^\omega \forall n[\alpha \circ \zeta (n)\ne 0]$, and that $D_\alpha$ is not-infinite if and only if $\lnot \exists \zeta \in [\omega ]^\omega \forall n[\alpha \circ \zeta (n)\ne 0]$. Decidable subsets of $\omega$ that are bounded-in-number are introduced and discussed in [34]. Almost-finite decidable subsets of $\omega$ were introduced in [34] and [35], and are also studied in [39].

Lemma 12.1

• (i) $\textsf{BIM}\vdash \forall \alpha [D_\alpha \; is\; finite\;\rightarrow D_\alpha \; is\;bounded$-in-number].

• (ii) $\textsf{BIM}\vdash \forall \alpha [D_\alpha \; is\; bounded$-in-$number\; \rightarrow \; D_\alpha \; is\; finite]\rightarrow \textbf{LPO}$.

• (iii) $\textsf{BIM}\vdash \forall \alpha [D_\alpha \; is\; bounded$-in-$number\rightarrow D_\alpha \; is\; almost$-finite].

• (iv) $\textsf{BIM}\vdash \forall \alpha [D_\alpha \; is\; almost$-$finite\; \rightarrow \;D_\alpha \; is\; bounded$-in-$number]\rightarrow \textbf{LPO}$.

• (v) $\textsf{BIM}\vdash \forall \alpha [D_\alpha \; is\; almost$-$finite\;\rightarrow \; D_\alpha \; is\; not$-infinite].

• (vi) $\textsf{BIM}+\textbf{BARIND}$²⁵$\vdash \forall \alpha [D_\alpha \; is\; almost$-$finite\;\rightarrow \; D_\alpha \; is\; not$-not-finite].

Proof

(i) Let $\alpha , n$ be given such that $\forall m\ge n[\alpha (m)=0]$. Note that $\forall t\in [\omega ]^{n+1}[t(n)\ge n]$ and conclude that $\forall t \in [\omega ]^{n+1}[\alpha \circ t(n)=0]$.

(ii) Assume $\forall \alpha [D_\alpha \; is\; bounded$-in-$number\; \rightarrow D_\alpha \; is\; finite]$. Let $\alpha$ be given. Define $\alpha ^*$ such that $\forall n[\alpha ^*(n)\ne 0\leftrightarrow n=\mu m[\alpha (m)\ne 0]]$. Note that $\forall t\in [\omega ]^2 \exists i<2[\alpha ^*\circ t(i)=0]$, so $D_{\alpha ^*}$ has at most one element and is bounded-in-number. Conclude that $D_{\alpha ^*}$ is finite and find n such that $\forall m\ge n[\alpha ^*(m) =0]$. Either $\exists m<n[\alpha ^*(m)\ne 0]$ or $\forall m[\alpha ^*(m)=0]$. Conclude that either $\exists m[\alpha (m)\ne 0]$ or $\forall m[\alpha (m)=0]$. We thus see that $\forall \alpha [\exists m[\alpha (m)\ne 0]\;\vee \;\forall m[\alpha (m)=0]]$, i.e. ${\textbf {LPO}}$.

(iii) Let $\alpha , n$ be given such that $\forall t \in [\omega ]^{n+1}\exists i<n+1[\alpha \circ t(i)=0]$. Conclude that $\forall \zeta \in [\omega ]^\omega \exists i < n+1[\alpha \circ \zeta (i)=0]$.

(iv) Assume $\forall \alpha [D_\alpha \; is\;almost$-$finite\; \rightarrow D_\alpha \; is\; bounded$-in-number]. Let $\alpha$ be given. Define $\alpha ^*$ such that $\forall n[\alpha ^*(n)\ne 0\leftrightarrow \mu m[\alpha (m)\ne 0]\le n<2\cdot \mu m[\alpha (m)\ne 0]]$. Note that, for all k, if $k=\mu [\alpha (m)\ne 0]$, then $\forall n[k\le n<2\cdot k\leftrightarrow \alpha ^*(n)\ne 0]$ and $\exists t \in [\omega ]^k\forall i<k[\alpha ^*\circ t(i)\ne 0]$ and $\forall t \in [\omega ]^{k+1}\exists i<k+1[\alpha ^*\circ t(i)= 0]$. Let $\zeta$ in $[\omega ]^\omega$ be given. We want to prove that $\exists n[\alpha ^*\circ \zeta (n)=0]$ and distinguish two cases. Case (a). $\alpha ^*\circ \zeta (0)= 0$. Then we are done. Case (b). $\alpha ^*\circ \zeta (0) \ne 0$. Then $\exists m[\alpha (m)\ne 0]$. Define $k:=\mu m[\alpha (m)\ne 0]$ and note: $\forall m\ge 2\cdot k[\alpha ^*(m)= 0]$, and, in particular, $\alpha ^*\circ \zeta (2\cdot k) =0$. Conclude that $\forall \zeta \in [\omega ]^\omega \exists n[\alpha ^*\circ \zeta (n)=0]$, i.e. $D_{\alpha ^*}$ is almost-finite. Using the assumption, conclude that $D_{\alpha ^*}$ is bounded-in-number. Find n such that $\forall t \in [\omega ]^{n+1}\exists i <n+1[\alpha ^*\circ t(i) =0]$. Conclude that, for all k, if $k=\mu m[\alpha (m)\ne 0]$, then $k<n+1$. Either $\exists k<n+1[\alpha (k)\ne 0]$ or $\forall k<n+1[\alpha (k)=0]$, and, therefore, either $\exists k[\alpha (k)\ne 0]$ or $\forall k[\alpha (k)=0]$. We thus see that $\forall \alpha [\exists k[\alpha (k)\ne 0]\;\vee \;\forall k[\alpha (k)=0]]$, i.e. ${\textbf {LPO}}$.

(v) The proof is left to the reader.

(vi) Let $\alpha$ be given such that $D_\alpha$ is almost-finite, i.e. $\forall \zeta \in [\omega ]^\omega \exists n[\alpha \circ \zeta (n)=0]$. Define $B:=\bigcup _n\{s\in \omega ^n\mid s\notin [\omega ]^n\;\vee \;\exists i<n[\alpha \circ s(i)=0]\}$. We now prove that B is a bar in $\omega ^\omega$. Let $\gamma$ be given. Define $\gamma ^*$ such that $\gamma ^*(0)=\gamma (0)$, and, for each n, if ${\overline{\gamma }}(n+2)\in [\omega ]^{<\omega }$, then $\gamma ^*(n+1)=\gamma (n+1)$, and, if not, then $\gamma ^*(n+1)=\gamma ^*(n)+1$. Note: $\gamma ^*\in [\omega ]^\omega$ and find n such that $\overline{\gamma ^*}n \in B$. Either ${\overline{\gamma }} n = \overline{\gamma ^*}n$ or ${\overline{\gamma }} n\notin [\omega ]^n$, and, in both cases, ${\overline{\gamma }} n \in B$. We thus see that $\forall \gamma \exists n[{\overline{\gamma }} n \in B]$ i.e. $Bar_{\omega ^\omega }(B)$. Define $E:=\bigcup _n\{s\in \omega ^n\mid s\notin [\omega ]^n \;\vee \; \exists i<n[\alpha \circ s(i)=0]\;\vee \;D_\alpha \;is\;not$-not-$finite]\}$.

Note that $B\subseteq E$.

We now prove that E is inductive. Let s, n be given such that $s\in \omega ^n$ and $\forall m[s*\langle m\rangle \in E]$. We have to prove that $s\in E$. We may assume that $s\in [\omega ]^n \;\wedge \;\lnot \exists i<n[\alpha \circ s(i)=0]$, and first consider two special cases. Case (a). $\exists m[s*\langle m \rangle \in [\omega ]^{n+1} \;\wedge \;\alpha (m)\ne 0]$. Finding such m, we consider $s*\langle m \rangle$ and conclude that $s*\langle m \rangle \in E$ and $D_\alpha$ is not-not-finite. Case (b). $\lnot \exists m[s*\langle m \rangle \in [\omega ]^{n+1} \;\wedge \;\alpha (m)\ne 0]$. Conclude that $\forall m[s*\langle m\rangle \in [\omega ]^{n+1}\rightarrow \alpha (m)=0]$ and that $D_\alpha$ is finite. Defining $P:=\exists m[s*\langle m \rangle \in [\omega ]^{n+1} \;\wedge \;\alpha (m)\ne 0]$, we may conclude that $(P\;\vee \;\lnot P)\rightarrow D_\alpha$ is not-not-finite. By intuitionistic logic,²⁶ we conclude that $D_\alpha$ is not-not-finite and $s\in E$. We thus see that $\forall s[\forall m[s*\langle m \rangle \in E]\rightarrow s\in E]$, i.e. E is inductive.

Obviously, E is monotone, i.e. $\forall s\forall m[s\in E\rightarrow s*\langle m \rangle \in E]$.

Using ${\textbf {BARIND}}$, we conclude that $\langle \;\rangle \in E$ and $D_\alpha$ is not-not-finite.

We thus see that $\textsf{BIM}+{\textbf {BARIND}}\vdash \forall \alpha [D_\alpha \; is\; almost$-$finite\;\rightarrow \; D_\alpha \; is\; not$-not-finite]. $\square$

Lemma 12.2

$\textsf{BIM}+\textbf{MP}$²⁷ proves $\forall \alpha [D_\alpha \; is\; not$-$infinite\;\leftrightarrow \;D_\alpha \; is\;not$-not-$finite\;\leftrightarrow D_\alpha \;is\;almost$-finite].

Proof

The proof is left to the reader. $\square$

We now extend the notion ‘almost-finite’ from decidable subsets of $\omega$ to enumerable subsets of $\omega$. For every $\alpha$, $E_\alpha :=\{n\mid \exists m[\alpha (m)=n+1]\}$, is the subset of $\omega$ enumerated by $\alpha$. We define: $E_\alpha$ is almost-finite if and only if $\forall \zeta \in [\omega ]^\omega \exists m\exists n[m<n\;\wedge \;\alpha \circ \zeta (m)=\alpha \circ \zeta (n)]$.

The first item of the next Lemma shows that the definition is a good definition indeed as it does not depend on the enumeration $\alpha$ of $E_\alpha$. The second item shows that this definition is consistent with the definition given earlier for decidable subsets of $\omega$. The fifth item shows that an almost-finite union of almost-finite enumerable subsets of $\omega$ is enumerable and almost-finite.

Lemma 12.3

$\textsf{BIM}$ proves the following.

• (i) $\forall \alpha \forall \beta [\bigl (E_\beta \subseteq E_\alpha \;\wedge \; \forall \zeta \in [\omega ]^\omega \exists m\exists n[m<n\;\wedge \;\alpha \circ \zeta (m)=\alpha \circ \zeta (n)]\bigr )\rightarrow \forall \zeta \in [\omega ]^\omega \exists m\exists n[m<n\;\wedge \;\beta \circ \zeta (m)=\beta \circ \zeta (n)]]$.

• (ii) $\forall \alpha \forall \beta [D_\alpha =E_\beta \rightarrow \bigl (\forall \zeta \in [\omega ]^\omega \exists n[\alpha \circ \zeta (n)=0]\leftrightarrow \forall \zeta \in [\omega ]^\omega \exists m\exists n[m<n\;\wedge \;\beta \circ \zeta (m)=\beta \circ \zeta (n)]\bigr )$.

• (iii) $\forall \alpha [\forall i<2[E_{\alpha ^{\upharpoonright i}}\;is\;almost$-$finite]\rightarrow \bigcup _{i<2}E_{\alpha ^{\upharpoonright i}}\; is\; almost$-finite].

• (iv) $\forall \alpha \forall n[ \forall i<n[E_{\alpha ^{\upharpoonright i}}\;is\;almost$-$finite]\rightarrow \bigcup _{i<n}E_{\alpha ^{\upharpoonright i}}\; is\; almost$-finite].

• (v) $\forall \alpha [(\forall n[E_{\alpha ^{\upharpoonright n}}\;is\;almost$-$finite]\;\wedge \;\forall \zeta \in [\omega ]^\omega \exists n[\alpha ^{\upharpoonright \zeta (n)}=\underline{0}])\rightarrow \bigcup _n E_{\alpha ^{\upharpoonright n}}$ $is\; almost$-finite].

Proof

(i) Let $\alpha , \beta$ be given such that $E_\beta \subseteq E_\alpha$ and $\forall \zeta \in [\omega ]^\omega \exists m \exists n[m<n \;\wedge \;\alpha \circ \zeta (m)=\alpha \circ \zeta (n)]$. Let $\zeta$ in $[\omega ]^\omega$ be given. We will prove that $\exists m \exists n[m<n \;\wedge \;\beta \circ \zeta (m)=\beta \circ \zeta (n)]$. Define $\zeta ^*$ such that $\zeta ^*(0)=\zeta (0)$ and, for all n, if $\forall i\le n+1[\beta \circ \zeta (i) \ne 0]$ and $\forall i<n+1\forall j\le n+1[i<j\rightarrow \beta (i)\ne \beta (j)]$, then $\zeta ^*(n+1) = \mu k[\alpha (k)=\beta \circ \zeta (n+1)]$, and, if not then $\zeta ^*(n+1) =\max _{i\le n} \zeta ^*(i)+1$. Note that $\forall m \forall n[m<n\rightarrow \zeta ^*(m+1)\ne \zeta ^*(n+1)]$. Find $\eta$ in $[\omega ]^\omega$ such that $\forall n[\zeta ^*\circ \eta (n+1)>\zeta ^*\circ \eta (n)]$. Find m, n such that $m<n$ and $\alpha \circ \zeta ^*\circ \eta (m)=\alpha \circ \zeta ^*\circ \eta (n)$. Either $\exists i\le \eta (n)[\beta \circ \zeta (i)=0]$ or $\beta \circ \zeta \circ \eta (m)=\alpha \circ \zeta ^*\circ \eta (m)=\alpha \circ \zeta ^*\circ \eta (n)=\beta \circ \zeta \circ \eta (n)$. We thus see that $\forall \zeta \in [\omega ]^\omega \exists m \exists n[m<n \;\wedge \;\bigl (\beta \circ \zeta (m) =0 \;\vee \; \beta \circ \zeta (m)=\beta \circ \zeta (n)\bigr )]$. One easily concludes that $\forall \zeta \in [\omega ]^\omega \exists m \exists n[m<n \;\wedge \;\beta \circ \zeta (m)=\beta \circ \zeta (n)]$

(ii) Let $\alpha , \beta$ be given such that $D_\alpha =E_\beta$. Define $\gamma$ as follows. For each n, if $\alpha (n)=0$, then $\gamma (n)=0$ and, if $\alpha (n)\ne 0$, then $\gamma (n)=\alpha (n)+1$. Note that $D_\alpha =E_\gamma$. $\gamma$ might be called the canonical enumeration of $D_\alpha$.

Assume $\forall \zeta \in [\omega ]^\omega \exists n[\alpha \circ \zeta (n)=0]$. Let $\zeta$ in $[\omega ]^\omega$ be given. Find m, n such that $m<n$ and $\alpha \circ \zeta (m) =\alpha \circ \zeta (n)=0$. Conclude that $\gamma \circ \zeta (m)=\gamma \circ \zeta (n)=0$. We thus see that $\forall \zeta \in [\omega ]^\omega \exists m \exists n[m<n \;\wedge \; \gamma \circ \zeta (m)=\gamma \circ \zeta (n)]$. Use (i) and conclude that $\forall \zeta \in [\omega ]^\omega \exists m \exists n[m<n \;\wedge \; \beta \circ \zeta (m)=\beta \circ \zeta (n)]$.

Now assume $\forall \zeta \in [\omega ]^\omega \exists m \exists n[m<n \;\wedge \;\beta \circ \zeta (m)=\beta \circ \zeta (n)]$. Use (i) and conclude that $\forall \zeta \in [\omega ]^\omega \exists m \exists n[m<n \;\wedge \;\gamma \circ \zeta (m)=\gamma \circ \zeta (n)]$. We will prove that $\forall \zeta \in [\omega ]^\omega \exists n[\alpha \circ \zeta (n)= 0]$. Let $\zeta$ in $[\omega ]^\omega$ be given. If $\alpha \circ \zeta (0) =0$ we are done. Now assume $\alpha \circ \zeta (0) \ne 0$. Define $\zeta ^*$ such that $\zeta ^*(0)=0$ and, for each n, if $\forall i\le n+1[\alpha \circ \zeta (i)\ne 0]$, then $\zeta ^*(n+1)=\zeta (n+1)$, and, if not, then $\zeta ^*(n+1)=\zeta ^*(n)+1$. Note that $\zeta ^*\in [\omega ]^\omega$ and find m, n such that $m<n$ and $\gamma \circ \zeta ^*(m)= \gamma \circ \zeta ^*(n)$. Conclude that $\exists i\le n[\alpha \circ \zeta (n)=0]$. We thus see that $\forall \zeta \in [\omega ]^\omega \exists n[\alpha \circ \zeta (n)= 0]$.

(iii) Let $\alpha$ be given such that $E_{\alpha ^{\upharpoonright 0}}, E_{\alpha ^{\upharpoonright 1}}$ are almost-finite. Define $\alpha ^*$ such that, for each n, $\alpha ^*(2n)=\alpha ^{\upharpoonright 0}(n)$ and $\alpha ^*(2n+1)=\alpha ^{\upharpoonright 1}(n)$ and note that $E_{\alpha ^*}=E_{\alpha ^{\upharpoonright 0}}\cup E_{\alpha ^{\upharpoonright 1}}$. Let $\zeta$ in $[\omega ]^\omega$ be given. We will prove $QED:=\exists m\exists n[m<n\;\wedge \;\alpha ^*\circ \zeta (m)=\alpha ^*\circ \zeta (n)]$. We first prove that $\forall k\exists l> k[\exists p[\zeta (l)=2p+1]\;\vee \;QED]$. Let k be given. Define $\zeta ^*$ such that, for each i, if $\forall j\le i\exists p[\zeta (k+1+j)=2p]$, then $\zeta ^*(i)=\zeta (k+1+i)$, and, if not, then $\zeta ^*(i)=2\cdot \zeta (k+1+i)$. Note that $\forall i\exists p[\zeta ^*(i)=2p]$. Define $\zeta ^{**}$ such that $\forall i[\zeta ^*(i)=2\cdot \zeta ^{**}(i)]$ and note that $\forall i[\alpha ^*\circ \zeta ^*(i)=\alpha ^0\circ \zeta ^{**}(i)]$. Find m, n such that $m<n$ and $\alpha ^0\circ \zeta ^{**}(m)=\alpha ^0\circ \zeta ^{**}(n)$ and note that either $\zeta ^*(m)=\zeta (k+1+m)$ and $\zeta ^*(n)=\zeta (k+1+n)$ and QED, or $\exists j\le n\exists p[\zeta (k+1+j)=2p+1]$. Using $\varvec{\Sigma }^0_1$-${\textbf {AC}}_{\omega ,\omega }$, see Theorem 4.1, find $\delta$ such that $\forall k[\delta (k)> k \;\wedge \; (\exists p[\zeta \circ \delta (k)=2p+1]\;\vee \;QED)]$. Define $\zeta ^\dag$ such that $\zeta ^\dag (0)=\delta (0)$ and, for each k, $\zeta ^\dag (k+1)=\delta \bigl (\zeta ^\dag (k)\bigr )$. Note that, for each k, if $\exists p[\zeta ^\dag (k)=2p]$, then QED. Define $\zeta ^{\dag *}$ such that, for each i, if $\forall j\le i\exists p[\zeta ^\dag (j)=2p+1]$, then $\zeta ^{\dag *}(i)=\zeta ^\dag (i)$, and, if not, then $\zeta ^{\dag *}(i)=2\cdot \zeta ^\dag (i)+1$. Note that $\forall i\exists p[\zeta ^{\dag ^*}(i)=2p+1]$. Define $\zeta ^{\dag **}$ such that $\forall i[\zeta ^{\dag *}(i)=2\cdot \zeta ^{\dag **}(i)+1]$ and note: $\forall i[\alpha ^*\circ \zeta ^{\dag *}(i)=\alpha ^1\circ \zeta ^{\dag **}(i)]$. Find m, n such that $m<n$ and $\alpha ^1\circ \zeta ^{\dag **}(m)=\alpha ^1\circ \zeta ^{\dag **}(n)$ and note that either $\zeta ^{\dag *}(m)=\zeta ^\dag (m)$ and $\zeta ^{\dag *}(n)=\zeta ^\dag (n)$ and QED, or $\exists j\exists p[\zeta ^\dag (j)=2p]$ and QED, so in any case QED. We thus see that $\forall \zeta \in [\omega ]^\omega \exists m\exists n[m<n\;\wedge \;\alpha ^*\circ \zeta (m)=\alpha ^*\circ \zeta (n)]$, i.e. $E_{\alpha ^*}= E_{\alpha ^{\upharpoonright 0}}\cup E_{\alpha ^{\upharpoonright 1}}$ is almost-finite.

(iv) Use (iii) and induction.

(v) Let $\alpha$ be given such that, for all n, $E_{\alpha ^{\upharpoonright n}}$ is almost-finite, and

$\forall \zeta \in [\omega ]^\omega \exists n[\alpha ^{\upharpoonright \zeta (n)}=\underline{0}]$. We will prove that $\bigcup _n E_{\alpha ^{\upharpoonright n}}$ is almost-finite. Define $\alpha ^*$ such that, for all p, $\alpha ^*(p)=\alpha ^{\upharpoonright p'}(p'')$ and note that $E_{\alpha ^*}= \bigcup _n E_{\alpha ^{\upharpoonright n}}$. Let $\zeta$ in $[\omega ]^\omega$ be given. We will prove $QED:=\exists m\exists n[m<n\;\wedge \;\alpha ^*\circ \zeta (m)=\alpha ^*\circ \zeta (n)]$. We first prove that $\forall k\forall n\exists l[l>k \;\wedge \;\bigl (QED\;\vee \;\zeta '(l)>n\bigr )]$. Let k, n be given. If $\zeta '(k+1)>n$, there is nothing to prove. Assume $\zeta '(k+1)\le n$. Define $\zeta ^*$ such that $\zeta ^*(0)=\zeta (k+1)$ and, for all i, if $\forall j\le i+1[\zeta '(k+1+j)\le n]$, then $\zeta ^*(i+1)=\zeta (k+2+i)$, and, if not, then $\zeta ^*(i+1)=\mu p[p>\zeta ^*(i)\;\wedge \;p'=n]$. Note that $\forall i[(\zeta ')*(i) \le n]$. Using (iii), find p, q such that $p<q$ and $\alpha ^*\circ \zeta ^*(p)=\alpha ^*\circ \zeta ^*(q)$ and note that either $\zeta ^*(p)=\zeta (k+1+p)$ and $\zeta ^*(q)=\zeta (k+1+q)$ and QED, or $\exists j\le q[\zeta '(k+1+j)>n]$. We thus see that $\forall k\forall n\exists l[l>k \;\wedge \;\bigl (QED\;\vee \;\zeta '(l)>n\bigr )]$. Using $\varvec{\Sigma }^0_1$-${\textbf {AC}}_{\omega ,\omega }$, see Theorem 4.1, find $\delta$ such that $\forall k\forall n[\delta (k,n)>k\;\wedge \;\big (QED\;\vee \;\zeta '\circ \delta (k,n)>n\bigr )]$. Define $\eta$ such that $\eta (0)=\delta (0,0)$, and, for each n, $\eta (n+1)=\delta \bigl (\eta (n),n\bigr )$. Note that $\eta \in [\omega ]^\omega$ and $\forall n[\zeta '\circ \eta (n)>n \;\vee \; QED]$. Find $\rho$ in $[\omega ]^\omega$ such that $\forall n[\zeta '\circ \eta \circ \rho (n+1)>\zeta '\circ \eta \circ \rho (n) \;\vee \;QED]$. Find p such that $\alpha ^{\upharpoonright \zeta '\circ \eta \circ \rho (p)}=\underline{0}$. Conclude that either QED, or $\alpha ^*\circ \zeta \circ \eta \circ \rho (\langle p,0\rangle ) = \alpha ^*\circ \zeta \circ \eta \circ \rho (\langle p,1\rangle )=0$, and again QED. We thus see that $\forall \zeta \in [\omega ]^\omega \exists m\exists n[m<n\;\wedge \;\alpha ^*\circ \zeta (m)=\alpha ^*\circ \zeta (n)]$, i.e. $E_{\alpha ^*}=\bigcup _n E_{\alpha ^{\upharpoonright n}}$ is almost-finite.

$\square$

12.2 Almost-fans and approximate fans

Recall that, for each $\beta$, $\mathcal {F}_\beta :=\{\alpha \mid \forall n[\beta ({\overline{\alpha }} n)=0]\}$. Let $\beta$ be given. We define the following.

$\beta$ is an almost-fan-law, $Almfan(\beta )$, if and only if $Spr(\beta )$ and

$\forall s \forall \zeta \in [\omega ]^\omega \exists m[\beta \bigl (s*\langle \zeta (m)\rangle \bigr )\ne 0]$. If $\beta$ is an almost-fan-law, then $\mathcal {F}_\beta$ is an almost-fan.

$\beta$ is an approximate-fan-law, $Appfan(\beta )$, if and only if $Spr(\beta )$ and $\forall n\exists k\forall t \in [\omega ]^{k+1}\exists i\le k[t(i)\notin \omega ^n\;\vee \;\beta \bigl (t(i)\bigr )\ne 0]$. If $\beta$ is an approximate-fan-law, then $\mathcal {F}_\beta$ is an approximate fan,²⁸.

$\beta$ is an explicit approximate-fan-law, $Appfan^+(\beta )$, if and only if $Spr(\beta )$ and $\exists \gamma \forall n\forall t \in [\omega ]^{\gamma (n)+1}\exists i\le \gamma (n)[t(i)\notin \omega ^n\;\vee \;\beta \bigl (t(i)\bigr )\ne 0]$. If $\beta$ is an explicit approximate-fan-law, then $\mathcal {F}_\beta$ is an explicit approximate fan.

Note that $\textsf{BIM}+$ Weak-$\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega , \omega }\vdash \forall \beta [Appfan(\beta )\rightarrow Appfan^+(\beta )]$.

12.3. The Almost-fan Theorem as a Scheme, ${\textbf {ALMFAN}}$,

$$\begin{aligned}{} & {} \forall \beta [\bigl (Almfan(\beta )\;\wedge \;Bar_{\mathcal {F}_\beta }(B)\bigr )\rightarrow \\{} & {} \exists \alpha [E_\alpha \subseteq B\;\wedge \;E_\alpha \;is\;almost{\text {-}}finite\;\wedge \;Bar_{\mathcal {F}_\beta }(E_\alpha )]]. \end{aligned}$$

The following theorem may be compared to Theorem 2.4.

Theorem 12.4

$\textsf{BIM}+\textbf{BARIND}+\textbf{AC}_{\omega ,\omega ^\omega }\vdash \textbf{ALMFAN}$.²⁹

Proof

Let $\beta$ be given such that $Almfan(\beta )$ and $\beta (\langle \;\rangle )=0$.³⁰ Assume $Bar_{\mathcal {F}_\beta }(B)$. Define $B':=B\cup \{s\mid \beta (s)\ne 0\}$. In the proof of Theorem 2.4 we have seen how to prove that $Bar_{\omega ^\omega }(B')$.

Let E be the set of all s such that either $\beta (s)\ne 0$ or $\beta (s) = 0$ and $\exists \alpha [ E_\alpha \subseteq B \;\wedge \; E_\alpha \;is\;almost$-$finite\;\wedge \; Bar_{\mathcal {F}_\beta \cap s}(E_\alpha )]$.

We prove that $B\subseteq E$. For every s, if $\beta (s) =0$ and $s\in B$, define $\alpha$ such that $\forall n[\alpha (n)=s+1]$ and note that $\{s\}=E_\alpha \subseteq B$ and $E_\alpha$ is finite and $Bar_{\mathcal {F}_\beta \cap s}(E_\alpha )$.

We prove that E is inductive. Let s be given such that $\forall m[s*\langle m \rangle \in E]$. Using ${\textbf {AC}}_{\omega , \omega ^\omega }$, find $\alpha$ such that, for all m, if $\beta (s*\langle m \rangle )=0$, then $E_{\alpha ^m}\subseteq B$ and $E_{\alpha ^m}$ is almost-finite and $Bar_{\mathcal {F}_\beta \cap s*\langle m \rangle }(E_{\alpha ^m})$, and, if $\beta (s*\langle m \rangle )\ne 0$, then $\alpha ^m = \underline{0}$ and $E_{\alpha ^m}=\emptyset$. Note that $Almfan(\beta )$ and $\forall \zeta \in [\omega ]^\omega \exists m[\alpha ^{\zeta (m)}=\underline{0}]$. Use Lemma 12.3(v) and conclude that $\bigcup _m E_{\alpha ^m}$ is almost-finite. Note that $Bar_{\mathcal {F}_\beta \cap s}(\bigcup _n E_{\alpha ^n})$ and conclude that $s\in E$. We thus see that $\forall s[\forall m[s*\langle m \rangle \in E]\rightarrow s \in E]$, i.e. E is inductive.

Note that E is also monotone, i.e. $\forall s\forall m[s\in E\rightarrow s*\langle m \rangle \in E]$.

Using ${\textbf {BARIND}}$, conclude: $\langle \;\rangle \in E$, i.e. $\exists \alpha [ E_\alpha \subseteq B \;\wedge \; E_\alpha \;is\;almost$-$finite\;\wedge \; Bar_{\mathcal {F}_\beta }(E_\alpha )]$. $\square$

12.4. The Almost-fan Theorem, ${\textbf {AlmFT}}$:

$$\begin{aligned}\forall \beta [Almfan(\beta )\rightarrow & {} \forall \alpha [\bigl ( Thinbar_{\mathcal {F}_\beta }(D_\alpha ) \;\wedge \; \forall s\in D_\alpha [\beta (s) = 0]\bigr )\\{} & {} \rightarrow \forall \zeta \in [\omega ]^\omega \exists n[\zeta (n) \notin D_\alpha ]]],\end{aligned}$$

Theorem 12.5

$\textsf{BIM}+\textbf{ALMFAN}\vdash \textbf{AlmFT}$.

Proof

Let $\beta , \alpha$ be given such that $Appfan(\beta )$ and $Thinbar_{\mathcal {F}_\beta }(D_\alpha )$ and $\forall s \in D_\alpha [\beta (s)=0]$. Applying Theorem 12.4, find $\gamma$ such that $E_\gamma \subseteq D_\alpha$ and $Bar_{\mathcal {F}_\beta }(E_\gamma )$ and $E_{\gamma }$ is almost-finite. As $\forall s\in D_\alpha \forall t\in D_\alpha [s\sqsubseteq t\rightarrow s=t]$, conclude that $E_\gamma = D_\alpha$ and, by Lemma 12.3(ii), $\forall \zeta \in [\omega ]^\omega \exists n[\zeta (n)\notin D_\alpha ]$. $\square$

The Almost-fan Theorem occurs in [37] and [38].

12.5. The Approximate-fan Theorem, ${\textbf {AppFT}}$:

$$\begin{aligned}{} & {} \forall \beta \forall \alpha [\bigl (Appfan^+(\beta ) \;\wedge \; Thinbar_{\mathcal {F}_\beta }(D_\alpha ) \;\wedge \; \forall s \in D_\alpha [ \beta (s) = 0]\bigr )\rightarrow \\{} & {} \forall \zeta \in [\omega ]^\omega \exists n[\zeta (n) \notin D_\alpha ]]. \end{aligned}$$

Theorem 12.6

$\textsf{BIM}+\textbf{AlmFT}\vdash \textbf{AppFT}$.

Proof

Obvious, as every approximate fan is an almost-fan.³¹$\square$

Theorem 12.7

$\textsf{BIM}+\textbf{AppFT}\vdash \textbf{FT}$.

Proof

Assume ${\textbf {AppFT}}$. Using Theorem , we will prove ${\textbf {FT}}$. Let $\alpha$ be given such that $Thinbar_{2^\omega }(D_\alpha )$ and $D_\alpha \subseteq 2^{<\omega }$. Using ${\textbf {AppFT}}$, conclude that $D_{\alpha }$ is almost-finite. Define $\zeta$ in $[\omega ]^\omega$ such that, for each n, if $\lnot \forall \gamma \in 2^\omega \exists i<n[\zeta (i)\sqsubset \gamma ]$, then $\zeta (n)=\mu p[p\in D_{\alpha }\;\wedge \;\forall i<n[p\ne \zeta (i)]]$. Find p such that $\zeta (p)\notin D_{\alpha }$. Conclude that $\forall n>\zeta (p)[n\notin D_\alpha ]$. We thus see that $\forall \alpha [Thinbar_{2^\omega }(D_\alpha )\rightarrow \exists m\forall n>m[n \notin D_\alpha ]]$. Using Theorem , we conclude ${\textbf {FT}}$. $\square$

$\textsf{BIM}$ does not prove ${\textbf {FT}}\rightarrow {\textbf {AppFT}}$, see [45, Corollary 10.6].

In $\textsf{BIM}$, ${\textbf {AppFT}}$ has a number of important equivalents. Two of them are the intuitionistic Infinite Ramsey theorem and a contrapositive form of the Bolzano-Weierstrass Theorem, see [45, Theorem 11.2 and Corollary 9.8] and [48, Sections 7.5 and 7.6].

The relation between ${\textbf {FT}}$ and ${\textbf {AppFT}}$ in $\textsf{BIM}$ may be compared to the relation between ${\textbf {WKL}}$ and ${\textbf {KL}}$ in $\textsf{RCA}_0$. In the classical context of $\mathsf {RCA_0}$, one studies two extensions of ${\textbf {WKL}}$. The first one is Bounded König’s Lemma ${\textbf {BKL}}$, that (for a classical reader) would coincide with (a contraposition of) the second formulation of ${\textbf {FT}}$ 2.2.4. The second one is König’s Lemma ${\textbf {KL}}$, that, similarly, would coincide with ${\textbf {FT}}^+$ 2.2.5. ${\textbf {BKL}}$ is, in $\mathsf {RCA_0}$, equivalent to ${\textbf {WKL}}$, see [27, Lemma IV.1.4], just as, in $\textsf{BIM}$, the first and second formulation of ${\textbf {FT}}$ are equivalent. ${\textbf {KL}}$, on the other hand, is definitely stronger than ${\textbf {WKL}}$, as $\mathsf {RCA_0} +{\textbf {KL}}$ is equivalent to $\mathsf {ACA_0}$.

As we observed before, see Theorem 4.2, $\textsf{BIM}+$Weak-$\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }\vdash {\textbf {FT}}\leftrightarrow {\textbf {FT}}^+.$³² From a constructive point of view, the axiom Weak-$\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$ is weak indeed, as the antecedent is read constructively. Note that $\textsf{BIM}+ {\textbf {AC}}_{\omega ,\omega }!\vdash$ Weak-$\varvec{\Pi }^0_1$-${\textbf {AC}}_{\omega ,\omega }$. In a classical context, the rôle of a countable axiom of choice is very different from the rôle of such an axiom in a constructive context, see [9, Appendix 1]. It seems that ${\textbf {FT}}^+$ is too close to ${\textbf {FT}}$ for being a good candidate to play, in the intuitionistic context, the rôle played by ${\textbf {KL}}$ in the classical context. Note that our ‘axiom’ ${\textbf {AppFT}}$ is a possibly better candidate, as, for a classical reader, ${\textbf {AppFT}}$ is also indistinguishable from ${\textbf {KL}}$.

Some authors have called our ${\textbf {FT}}$ the Weak Fan Theorem ${\textbf {WFT}}$, see for instance [20], but we decided not to follow them. There is a constructive version of Weak Weak König’s Lemma ${\textbf {WWKL}}$, see [27, Definition X.1.7], that is called ${\textbf {WWFT}}$, see [24].

13 Notation and conventions

In this Section we explain how, in $\textsf{BIM}$, some useful notation is introduced and some elementary results are proven.

13.1 Finite and infinite sequences of natural numbers

$\textsf{BIM}$ contains a constant p denoting the function enumerating the prime numbers: $p(0)= 2, p(1)= 3, \ldots$

We code finite sequences of natural numbers by natural numbers:

$\langle \;\rangle :=0$ and, for each $k>0$, for all $m_0,m_1, \ldots m_{k-1}$,

$$\begin{aligned}\langle m_0, \ldots , m_{k-1} \rangle = p(k-1)\cdot \prod _{i<k}p(i)^{m_i}-1 \end{aligned}$$

$length(0):=0$ and,

for each $s>0$, $\textit{length}(s):= 1\;+$ the largest k such that p(k) divides $s+1$.

If $i<\textit{length}(s)-1$, then $s(i):=$ the largest m such that $p(i)^m$ divides $s+1$, and, if $i=\textit{length}(s)-1$, then $s(i):=$ the largest m such that $p(i)^{m+1}$ divides $s+1$, and, if $i\ge \textit{length}(s)$ then $s(i):=0$.

Note: if $\textit{length}(s) = k$, then $s =\langle s(0), s(1), \ldots , s(k-1)\rangle$.

Also note: $\forall s[s\ge \textit{length}(s)]$.

$a*b$ is the number s satisfying: $\textit{length}(s) = \textit{length}(a) + \textit{length}(b)$ and,

for each n, if $n < length(a)$, then $s(n) = a(n)$ and,

if $\textit{length}(a) \le n <\textit{length}(s)$, then $s(n) = b \bigl (n - \textit{length}(a)\bigr )$.

$a *\alpha$ is the element $\beta$ of $\omega ^\omega$ satisfying: for each n, if $n< \textit{length}(a)$, then $\beta (n) = a(n)$, and, if $\textit{length}(a)\le n$, then $\beta (n) = \alpha \bigl (n - \textit{length}(a)\bigr )$.

For $n \le length(a)$, ${\overline{a}}(n):= \langle a(0), \ldots , a(n-1) \rangle$.

If confusion seems unlikely, we sometimes write: “${\overline{a}} n$” and not: “${\overline{a}} (n)$”.

$a \sqsubseteq b\leftrightarrow \exists n \le \textit{length}(b)[a = {\overline{b}} n]$, and: $a \sqsubset b\leftrightarrow (a\sqsubseteq b\;\wedge \;a \ne b)$.

$s\le _{lex}t \leftrightarrow \forall m <\textit{length}(s)[{\overline{s}} m ={\overline{t}} m \rightarrow s(m)\le t(m)]$.

$a \perp b\leftrightarrow \lnot (a\sqsubseteq b\;\vee \;b \sqsubseteq a)$.

$\overline{\alpha }(n):={\overline{\alpha }} n:= \langle \alpha (0), \ldots \alpha (n-1) \rangle$.

$a \sqsubset \alpha \leftrightarrow \exists n[{\overline{\alpha }} n = a]$.

$a\perp \alpha \leftrightarrow \alpha \perp a\leftrightarrow \lnot (a\sqsubset \alpha )$.

$\alpha \;\#\;\beta \leftrightarrow \alpha \perp \beta \leftrightarrow \exists n[\alpha (n)\ne \beta (n)]$.

For each p, $\underline{p}$ is the element of $\omega ^\omega$ satisfying $\forall n[\underline{p} (n) =p]$.

$\forall \alpha \forall n[ \alpha '(n)=\bigl (\alpha (n)\bigr )'\;\wedge \; \alpha ''(n)=\bigl (\alpha (n)\bigr )'']$.

We extend the language of $\textsf{BIM}$ by introducing $\in$ and terms denoting subsets of $\omega$. This is not a real extension of the language of $\textsf{BIM}$. Formulas in which the new symbols occur are abbreviations of formulas in which they do not occur.

Given a formula $\varphi =\varphi (n)$ we may introduce a ‘set term’ $T_\varphi$. The basic formula ‘$t \in X_\varphi$’ is an abbreviation of ‘$\varphi (t)$’.

Here is a first example.

$2^{<\omega }:=\textit{Bin}:=\{a\mid \forall n < \textit{length}(a)[a(n) = 0 \; \vee \; a(n) = 1]\}$.

‘$a\in 2^{<\omega }$’ is an abbreviation of ‘$\forall n < \textit{length}(a)[a(n) = 0 \; \vee \; a(n) = 1]$’.

$\omega ^m:=\{s\mid length(s)=m\}$.

$[\omega ]^m:=\{s \in \omega ^m\mid \forall i[i+1<m\rightarrow s(i)<s(i+1)]\}$.

Given terms A, B denoting subsets of $\omega$,

‘$A\subseteq B$’ is an abbreviation of ‘$\forall n[n\in A\rightarrow n \in B]$’.

We also introduce terms denoting subsets of $\omega ^\omega$, for instance:

$2^\omega :=\{\gamma \mid \forall n [ \gamma (n) = 0 \vee \gamma (n) = 1]\}$.

‘$\alpha \in 2^\omega$’ is an abbreviation of ‘$\forall n [\alpha (n) = 0 \; \vee \; \alpha (n) = 1]$’.

$[\omega ]^\omega :=\{\zeta \mid \forall n[\zeta (n)<\zeta (n+1)]\}$.

Given terms $\mathcal {X}, \mathcal {Y}$ denoting subsets of $\omega ^\omega$,

‘$\mathcal {X}\subseteq \mathcal {Y}$’ abbreviates ‘$\forall \alpha [\alpha \in \mathcal {X}\rightarrow \alpha \in \mathcal {Y}]$’.

Given a term $\mathcal {X}$ denoting a subset of $\omega ^\omega$, we introduce, for all s,

$\mathcal {X}\cap s:=\{\alpha \in \mathcal {X}\mid s\sqsubset \alpha \}$.

Given a term $\mathcal {X}$ denoting a subset of $\omega ^\omega$, and a term B denoting a subset of $\omega$, we let ‘$\textit{Bar}_\mathcal {X}(B)$’ be an abbreviation of ‘$\forall \alpha \in \mathcal {X}\exists n[{\overline{\alpha }} n \in B]$’.

Note that $\textit{Bar}_\mathcal {X}(B)$ is a formula scheme, that becomes a formula if one substitutes formulas defining $\mathcal {X}$, B, respectively.

From now on, we will express ourselves more informally, as in the following example:

For each $\mathcal {X}\subseteq \omega ^\omega$, for each $B\subseteq \omega$,

$\textit{Thinbar}_\mathcal {X}(B)\leftrightarrow \bigl (Bar_\mathcal {X}(B)\;\wedge \; \forall s\in B\forall t\in B[s\sqsubseteq t\rightarrow s=t]\bigr )$.

Note that we are not extending the language of $\textsf{BIM}$ by second-order variables.

13.2 Decidable and enumerable subsets of $\omega$

$D_\alpha :=\{i\mid \alpha (i)\ne 0\}$. $D_\alpha$ is the subset of $\omega$ decided by $\alpha$.

The expression ‘$i\in D_\alpha$’ is an abbreviation of ‘$\alpha (i)\ne 0$’.

$X\subseteq \omega$ is decidable or $\varvec{\Delta }^0_1$ if and only if $\exists \alpha [X=D_\alpha ]$.

$D_a:=\{i\mid i < \textit{length}(a)\mid a(i) \ne 0\}$.

$X\subseteq \omega$ is finite if and only if $\exists a[X=D_a]$.

Note: for each $\alpha$, $D_\alpha = \bigcup \nolimits _{n \in \omega } D_{{\overline{\alpha }} n}$.

$E_\alpha := \{n\mid \exists p [\alpha (p) = n+ 1] \}$. $E_\alpha$ is the subset of $\omega$ enumerated by $\alpha$.

$X\subseteq \omega$ is enumerable or $\varvec{\Sigma }^0_1$ if and only if $\exists \alpha [X=E_\alpha ]$.

$E_a:= \{n\mid \exists p < \textit{length}(a)[a(p) = n+ 1] \}$.

Note: for each $\alpha$, $E_\alpha = \bigcup \nolimits _{n \in \omega } E_{{\overline{\alpha }} n}$.

$X\subseteq \omega$ is co-enumerable or $\varvec{\Pi }^0_1$ if and only if $\exists \alpha [X=\omega {\setminus } E_\alpha =\{n\mid \forall p[\alpha (p)\ne n+1]\}]$.

Given any $\alpha$, define $\beta$ such that

$\forall n[\alpha (n)=\beta (n) = 0\;\vee \; \bigl (\alpha (n)\ne 0\;\wedge \;\beta (n)=n+1\bigr )]$, and note: $D_\alpha = E_\beta$.

We thus see: $\textsf{BIM}\vdash \forall \alpha \exists \beta [D_\alpha = E_\beta ]$.

13.3 Open and closed subsets of $\omega ^\omega$, spreads and fans

$\mathcal {G}_\beta :=\{\gamma \mid \exists n[\beta ({\overline{\gamma }} n) \ne 0]\}$.

$\mathcal {G}\subseteq \omega ^\omega$ is open or $\varvec{\Sigma }^0_1$ if and only if $\exists \beta [\mathcal {G}=\mathcal {G}_\beta ]$.

$\mathcal {F}_\beta := \omega ^\omega {\setminus } \mathcal {G}_\beta = \{\gamma \mid \forall n[\beta ({\overline{\gamma }} n) =0]\}$.

$\mathcal {F}\subseteq \omega ^\omega$ is closed or $\varvec{\Pi }^0_1$ if and only if $\exists \beta [\mathcal {F}= \mathcal {F}_\beta ]$.

$Spr(\beta )\leftrightarrow \forall s[\beta (s) =0\leftrightarrow \exists n[\beta (s*\langle n \rangle )=0]]$.

$\mathcal {X}\subseteq \omega ^\omega$ is a spread if and only if $\exists \beta [Spr(\beta )\;\wedge \;\mathcal {X}=\mathcal {F}_\beta ]$.

In intuitionistic mathematics, not every closed subset of $\omega ^\omega$ is a spread, see Lemma 2.12.

$Fan(\beta )\leftrightarrow \bigl (Spr(\beta )\;\wedge \;\forall s\exists n\forall m[\beta (s*\langle m \rangle )=0\rightarrow m\le n]\bigr )$ and

$Fan^+(\beta )\leftrightarrow \bigl (Spr(\beta )\;\wedge \;\exists \gamma \forall s\forall m[\beta (s*\langle m \rangle )=0\rightarrow m\le \gamma (s)]\bigr )$.

$\mathcal {X}\subseteq \omega ^\omega$ is a fan if and only if $\exists \beta [Fan(\beta )\;\wedge \;\mathcal {X}=\mathcal {F}_\beta ]$.

$\mathcal {X}\subseteq \omega ^\omega$ is an explicit fan if and only if $\exists \beta [Fan^+ (\beta )\;\wedge \;\mathcal {X}=\mathcal {F}_\beta ]$.

13.4 Subsequences

$\forall n\forall m[\alpha ^{\upharpoonright n}(m):= \alpha (\langle n \rangle *m)]$.

$\alpha ^{\upharpoonright n}$ is called the n-th subsequence of the infinite sequence $\alpha$.³³

$\textit{length}(s^{\upharpoonright n}):=\mu p\le s[\langle n \rangle *p \ge \textit{length}(s)]$ and

$\forall m<length(s^{\upharpoonright n})[s^{\upharpoonright n}(m) = s(\langle n \rangle *m)].$

Note that $\forall \alpha \forall m\forall n[({\overline{\alpha }} m)^{\upharpoonright n}\sqsubset \alpha ^{\upharpoonright n}]$.

$\forall m[^a \alpha (m):= \alpha (a *m)]$.

Note: $\forall n[^{\langle n\rangle }\alpha = \alpha ^{\upharpoonright n}]$.

$\textit{length}(^as):=\mu p\le s[a*p \ge \textit{length}(s)]$ and $\forall m<length(^as)[^as(m) = s(a *m)].$

Note: $\forall \alpha \forall m\forall a[^a({\overline{\alpha }} m)\sqsubset \;^a\alpha ]$.

13.5 Partial continuous functions from $\omega ^\omega$ to $\omega$ and from $\omega ^\omega$ to $\omega ^\omega$

$Fun_0(\varphi )\leftrightarrow \forall a \in E_\varphi \forall b \in E_\varphi [a'\sqsubseteq b'\rightarrow a'' = b'']$.³⁴

$Dom_0(\varphi ):=\{\alpha \mid \exists a \in E_\varphi [a'\sqsubset \alpha ]\}$.

Assume: $Fun_0(\varphi )$ and $\alpha \in Dom_0(\varphi )$.

Then $\varphi (\alpha ) :=\; the\;number\; c\; such \;that\; \exists n[({\overline{\alpha }} n, c)\in E_\varphi ]$.

For every $\mathcal {X}\subseteq \omega ^\omega$, for every $\varphi$, $\varphi :\mathcal {X}\rightarrow \omega \leftrightarrow \bigl (Fun_0(\varphi )\;\wedge \; \mathcal {X}\subseteq Dom_0(\varphi )\bigr )$.

$Fun_1(\varphi )\leftrightarrow \forall a \in E_\varphi \forall b \in E_\varphi [a'\sqsubseteq b'\rightarrow a'' \sqsubseteq b'']$.

$Dom_1(\varphi ):=\{\alpha \mid \forall n \exists a \in E_\varphi [a'\sqsubset \alpha \;\wedge \; length(a'')\ge n]\}$.

$\varphi |a:=\max (\{t\mid \exists b \in E_{{\overline{\varphi }} a}[b'\sqsubseteq a \;\wedge b''=t]\})$.

$\varphi :\alpha \mapsto \gamma \leftrightarrow \forall n\exists m[{\overline{\gamma }} n \sqsubseteq \varphi |{\overline{\alpha }} m]]$.

Assume: $Fun_1(\varphi )$ and $\alpha \in Dom_1(\varphi )$.

Then $\varphi |\alpha := \; the \; element \;\gamma \;of\; \omega ^\omega$ such that $\varphi :\alpha \mapsto \gamma$.

For every $\mathcal {X}\subseteq \omega ^\omega$, for every $\varphi$, $\varphi :\mathcal {X}\rightarrow \omega ^\omega \leftrightarrow \bigl (Fun_1(\varphi )\;\wedge \; \mathcal {X}\subseteq Dom_1(\varphi )\bigr )$.

13.6 Integers and rationals

$m=_\mathbb {Z}n\leftrightarrow m'+n''=m''+n'$.

$m<_\mathbb {Z}n\leftrightarrow m'+n''<m''+n'$.

$0_\mathbb {Z}:=(0,0)$

$m +_\mathbb {Z} n= (m'+n',m''+n'')$.

$m -_\mathbb {Z} n= (m'+n'',m''+n')$.

$m\cdot _\mathbb {Z} n:=(m'\cdot n'+m''\cdot n'', m'\cdot n''+m''\cdot n')$.

$\mathbb {Q}:=\{n\mid n''>_\mathbb {Z} 0_\mathbb {Z}\}$.

$m=_\mathbb {Q}n\leftrightarrow m'\cdot _\mathbb {Z}n''=_\mathbb {Z} m''\cdot _\mathbb {Z} n'$.

$m<_\mathbb {Q}n\leftrightarrow m'\cdot _\mathbb {Z}n''<_\mathbb {Z} m''\cdot _\mathbb {Z} n'$.

$m\le _\mathbb {Q} n\leftrightarrow m'\cdot _\mathbb {Z}n''\le _\mathbb {Z} m''\cdot _\mathbb {Z} n'$.

$m\le _\mathbb {Q}n \leftrightarrow \max _\mathbb {Q}(m,n) =_\mathbb {Q}n \leftrightarrow \min _\mathbb {Q}(m,n) =_\mathbb {Q}m$.

$m +_\mathbb {Q} n= (m'\cdot _\mathbb {Z}n''+_\mathbb {Z}m''\cdot _\mathbb {Z}n',m''\cdot _\mathbb {Z}n'')$.

$m -_\mathbb {Q} n= (m'\cdot _\mathbb {Z}n''-_\mathbb {Z}m''\cdot _\mathbb {Z} n',m''\cdot _\mathbb {Z}n'')$.

$m\cdot _\mathbb {Q} n=(m'\cdot _\mathbb {Z} n', m''\cdot _\mathbb {Z}n'')$.

$\mathbb {S}:=\{s\mid s'\in \mathbb {Q} \;\wedge \; s''\in \mathbb {Q}\;\wedge \;s'<_\mathbb {Q}s''\}$.

$s \sqsubset _\mathbb {S} t\leftrightarrow (t'<_\mathbb {Q} s' \;\wedge \; s'' <_\mathbb {Q} t'')$.

$s \sqsubseteq _\mathbb {S} t\leftrightarrow (t'\le _\mathbb {Q} s' \;\wedge \; s'' \le _\mathbb {Q} t'')$.

$s<_\mathbb {S} t\leftrightarrow s''<_\mathbb {Q} t'$.

$s\le _\mathbb {S} t\leftrightarrow s'\le _\mathbb {Q} t''$.

$s\;\#_\mathbb {S} \;t \leftrightarrow (s<_\mathbb {S} t \;\vee \; t<_\mathbb {S} s)$.

For each n, we define $n_\mathbb {Q}$ in $\mathbb {Q}$ by: $n_\mathbb {Q}=\bigl ((n,0),(1,0)\bigr )$.

For all s in $\mathbb {S}$, $\textit{double}_\mathbb {S}(s)$ is the element u of $\mathbb {S}$ satisfying:

$u'+_\mathbb {Q}u''=_\mathbb {Q} s'+_\mathbb {Q} s''$ and $u''-_\mathbb {Q} u' =_\mathbb {Q} 2_\mathbb {Q}\cdot _\mathbb {Q} (s''-_\mathbb {Q} s')$.

Note that $\forall s \in \mathbb {S}\forall t \in \mathbb {S}[s\sqsubseteq _\mathbb {S} t\rightarrow \textit{double}_\mathbb {S}(s) \sqsubset _\mathbb {S} \textit{double}_\mathbb {S}(t)]$.

$s+_\mathbb {S} t:= (s'+_\mathbb {Q} t', s''+_\mathbb {Q} t'')$.

$s\cdot _\mathbb {S} t:= \bigl (\min _\mathbb {Q}(s'\cdot _\mathbb {Q}t', s''\cdot _\mathbb {Q}t', s'\cdot _\mathbb {Q}t'', s''\cdot _\mathbb {Q}t''), \max _\mathbb {Q}(s'\cdot _\mathbb {Q}t', s''\cdot _\mathbb {Q}t', s'\cdot _\mathbb {Q}t'', s''\cdot _\mathbb {Q}t'')\bigr )$.

13.7 Real numbers

$\mathcal {R}:=\{ \alpha \mid \forall n [\alpha (n)\in \mathbb {S}\;\wedge \;\alpha (n+1) \sqsubseteq _\mathbb {S} \alpha (n)]\;\wedge \;\forall m \exists n[\alpha ''(n) -_\mathbb {Q} \alpha '(n) <_\mathbb {Q} \frac{1}{2^m}]\}$.³⁵

$\alpha<_\mathcal {R} \beta \leftrightarrow \exists n[\alpha (n) <_\mathbb {S} \beta (n)]$.

$\alpha \le _\mathcal {R} \beta \leftrightarrow \forall n[\alpha (n) \le _\mathbb {S} \beta (n)]$.

$\forall n[\inf (\alpha ,\beta )(n):=\bigl (\min _\mathbb {Q}(\alpha '(n),\beta '(n)),\min _\mathbb {Q}(\alpha ''(n),\beta ''(n))\bigr )]$.

$\forall n[\sup (\alpha ,\beta )(n):=\bigl (\max _\mathbb {Q}(\alpha '(n),\beta '(n)),\max _\mathbb {Q}(\alpha ''(n),\beta ''(n))\bigr )]$.

$\alpha \;\#_\mathcal {R}\; \beta \leftrightarrow (\alpha<_\mathcal {R} \beta \;\vee \; \beta <_\mathcal {R} \alpha )$.

$\alpha =_\mathcal {R} \beta \leftrightarrow (\alpha \le _\mathcal {R} \beta \;\wedge \;\beta \le _\mathcal {R} \alpha )$.

$\forall n[(\alpha +_\mathcal {R}\beta )(n):=\alpha (n)+_\mathbb {S}\beta (n)]$.

$\forall n[(\alpha \cdot _\mathcal {R}\beta )(n):=\alpha (n)\cdot _\mathbb {S}\beta (n)]$.

For each q in $\mathbb {Q}$, we define $q_\mathcal {R}$ in $\mathcal {R}$ by: for each n, $q_\mathcal {R}(n) = (q -_\mathbb {Q} \frac{1}{2^n}, q+_\mathbb {Q} \frac{1}{2^n})$.

$0_\mathcal {R}:=(0_\mathbb {Q})_\mathcal {R}$ and $1_\mathcal {R}:=(1_\mathbb {Q})_\mathcal {R}$.

Lemma 13.1

One may prove in $\textsf{BIM}$: $\forall s \in \mathbb {S} \forall t \in \mathbb {S}[s\sqsubset _\mathbb {S} t \rightarrow \forall \alpha \in \mathcal {R} \exists n[s\;\#_\mathbb {S} \;\alpha (n)\;\vee \;\alpha (n) \sqsubset _\mathbb {S} t]]$.

Proof

The proof is left to the reader. $\square$

13.8 [0, 1] and $2^\omega$

$[0,1]:=\{\alpha \in \mathcal {R}\mid 0_\mathcal {R} \le _\mathcal {R} \alpha \le _\mathcal {R} 1_\mathcal {R}\}$.

$(0,1]:=\{\alpha \in \mathcal {R}\mid 0_\mathcal {R} <_\mathcal {R} \alpha \le _\mathcal {R} 1_\mathcal {R}\}$, and $[0,1):=\{\alpha \in \mathcal {R}\mid 0_\mathcal {R} \le _\mathcal {R} \alpha <_\mathcal {R} 1_\mathcal {R}\}$.

For all $\alpha , \beta$ in $\mathcal {R}$, $[\alpha ,\beta ):=\{\gamma \in \mathcal {R}\mid \alpha \le _\mathcal {R}\gamma <_\mathcal {R}\beta \}$.

$[0,1]^2:=\{\gamma \mid \forall i<2[\gamma ^{\upharpoonright i}\in [0,1]]\}$ and $[0,1]^\omega :=\{\gamma \mid \forall n[\gamma ^{\upharpoonright n}\in [0,1]]\}$.

$\mathcal {H}_\alpha :=\{\gamma \in [0,1]\mid \exists n\exists s \in \mathbb {S}[\alpha (s) \ne 0 \;\wedge \; \gamma (n)\sqsubset _\mathbb {S} s ]\}$.

$\mathcal {H}\subseteq \mathcal {R}$ is open if and only if $\exists \alpha [\mathcal {H}=\mathcal {H}_\alpha ]$.

Lemma 13.2

One may prove in $\textsf{BIM}$:

There exist $\sigma , \psi$ such that

• (i) $\sigma :2^\omega \rightarrow [0,1]$ and $\forall \delta \in [0,1]\exists \gamma \in 2^\omega [\sigma |\gamma =_\mathcal {R}\delta ]$.

• (ii) $\psi :\omega ^\omega \rightarrow \omega ^\omega$ and $\forall \alpha \forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_{\psi |\alpha }\leftrightarrow \sigma |\gamma \in \mathcal {H}_\alpha ]$.

Proof

(i) Define $\lambda$ and $\rho$ such that, for each s in $\mathbb {S}$, $\lambda (s)=( s', \frac{1}{3}s'+_\mathbb {Q}\frac{2}{3}s'')$ and $\rho (s)=( \frac{2}{3}s'+_\mathbb {Q}\frac{1}{3}s'', s'')$. For each s in $\mathbb {S}$, $\lambda (s)$ is the left-two-third part of s and $\rho (s)$ is the right-two-third part of s. Define $\nu$ such that $\nu (\langle \;\rangle )=(0_\mathbb {Q}, 1_\mathbb {Q})$ and, for all s in Bin, $\nu (s*\langle 0\rangle )=\lambda \bigl (\nu (s)\bigr )$ and $\nu (s*\langle 1\rangle )=\rho \bigl (\nu (s)\bigr )$. Define $\sigma :2^\omega \rightarrow [0,1]$ such that $\forall \gamma \in 2^\omega [(\sigma |\gamma )(n)=\nu ({\overline{\gamma }} n)]$. One may prove that $\forall \delta \in [0,1]\exists \gamma \in 2^\omega [\sigma |\gamma =_\mathcal {R}\delta ]$.

(ii) Define $\psi :\omega ^\omega \rightarrow \omega ^\omega$ such that $\forall \alpha \forall s[(\psi |\alpha )(s)\ne 0 \leftrightarrow \bigl (s\in 2^{<\omega }\;\wedge \;\exists t<s[\nu (s)\sqsubset _\mathbb {S}t \;\wedge \;\alpha (t)\ne 0]\bigr )]$. One may prove that $\forall \alpha \forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_{\psi |\alpha }\leftrightarrow \sigma |\gamma \in \mathcal {H}_\alpha ]$.$\square$

Lemma 13.3

One may prove in $\textsf{BIM}$:

There exist $\tau , \chi$ such that

• (i) $\tau :2^\omega \rightarrow [0,1]$ and $\forall \gamma \in 2^\omega \forall \delta \in 2^\omega [\gamma \;\#\;\delta \rightarrow \tau |\gamma \;\#_\mathcal {R}\;\tau |\delta ]$.

• (ii) $\chi :\omega ^\omega \rightarrow \omega ^\omega$ and $\forall \alpha \forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_\alpha \leftrightarrow \tau |\gamma \in \mathcal {H}_{\chi |\alpha }]$.

• (iii) $\forall \delta \in [0,1]\exists \gamma \in 2^\omega [\tau |\gamma \;\#_\mathcal {R}\;\delta \rightarrow \forall \alpha [\delta \in \mathcal {H}_{\chi |\alpha }]]$.

• (iv) $\forall \delta \in [0,1]^\omega \exists \gamma \in 2^\omega \forall n[\tau |\gamma ^{\upharpoonright n}\;\#_\mathcal {R}\;\delta ^{\upharpoonright n}\rightarrow \forall \alpha [\delta ^{\upharpoonright n} \in \mathcal {H}_{\chi |\alpha }]]$.

Proof

(i) Define $\pi _0, \pi _1, \pi _2, \pi _3$ and $\pi _4$ such that, for each s in $\mathbb {S}$, for each $i<5$, $\pi _i(s):=( \frac{5-i}{5}s' +_\mathbb {Q} \frac{i}{5} s'',\frac{5-i-1}{5}s' +_\mathbb {Q} \frac{i+1}{5} s'')$. For each s in $\mathbb {S}$, for each $i<5$, $\pi _i(s)$ is the i-th fifth part of s. Define $\varepsilon$ such that $\varepsilon (\langle \;\rangle )=(0_\mathbb {Q}, 1_\mathbb {Q})$ and, for all a in $2^{<\omega }$, $\varepsilon (a*\langle 0\rangle )=\pi _1\bigl (\varepsilon (a)\bigr )$ and $\varepsilon (s*\langle 1\rangle )=\pi _3\bigl (\varepsilon (a)\bigr )$. Define $\tau :2^\omega \rightarrow [0,1]$ such that $\forall \gamma \in 2^\omega [(\tau |\gamma )(n)=\varepsilon ({\overline{\gamma }} n)]$. One may prove that $\forall \gamma \in 2^\omega \forall \delta \in 2^\omega [\gamma \;\#\;\delta \rightarrow \tau |\gamma \;\#_\mathcal {R}\;\tau |\delta ]$.

(ii) Define $\chi :\omega ^\omega \rightarrow \omega ^\omega$ such that, for all $\alpha$, for all s, $(\chi |\alpha )(s)\ne 0$ if and only if $\exists t<s[ t\in 2^{<\omega }\;\wedge \; s\sqsubset _\mathbb {S} \varepsilon (t)\;\wedge \;\alpha (t)\ne 0]\;\vee \;\exists n<s\forall t\in 2^{<\omega }[length(t)=n\rightarrow s\;\#_\mathbb {S}\;\varepsilon (t)]$. We now prove that $\forall \alpha \forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_{\alpha }\leftrightarrow \tau |\gamma \in \mathcal {H}_{\chi |\alpha }]$ Let $\alpha , \gamma$ be given such that $\gamma \in 2^\omega$. Assume that $\gamma \in \mathcal {G}_\alpha$. Find n such that $\alpha ({\overline{\gamma }} n)\ne 0$. Find $k>n$ such that $\varepsilon ({\overline{\gamma }} k) > {\overline{\gamma }} n$. Note that $\varepsilon ({\overline{\gamma }} k) \sqsubset _\mathbb {S} \varepsilon ({\overline{\gamma }} n)$ and conclude that$(\chi |\alpha )\bigl (\varepsilon ({\overline{\gamma }} k)\bigr )\ne 0$. As $(\tau |\gamma )(k+1) \sqsubset _\mathbb {S} (\tau |\gamma )(k)=\varepsilon ({\overline{\gamma }} k)$, conclude that $\tau |\gamma \in \mathcal {H}_{\chi |\alpha }$. Conversely, assume that $\tau |\gamma \in \mathcal {H}_{\chi |\alpha }$. Note that $\forall n[\varepsilon ({\overline{\gamma }} n) \sqsubset \tau |\gamma ]$. Conclude that$\lnot \exists s[s\sqsubset \tau |\gamma \;\wedge \;\exists n\forall t\in 2^{<\omega }[length(t)=n\rightarrow s\;\#_\mathbb {S}\;\varepsilon (t)]]$. Find n, s such that $(\tau |\gamma )(n)\sqsubset _\mathbb {S} s$ and $(\chi |\alpha )(s)\ne 0$. Find t in $2^{<\omega }$ such that $s\sqsubset \varepsilon (t)$ and $\alpha (t)\ne 0$. Note that $t\sqsubset \gamma$ and conclude that $\gamma \in \mathcal {G}_\alpha$. We thus see that $\forall \gamma \in 2^\omega [\gamma \in \mathcal {G}_\alpha \leftrightarrow \tau |\gamma \in \mathcal {H}_{\chi |\alpha }]$.

(iii) Assume that $\delta \in [0,1]$. Note that $\forall a \in 2^{<\omega }[\varepsilon ''(a*\langle 0\rangle )<_\mathbb {Q}\varepsilon '(a*\langle 1\rangle )]$. Define $\gamma$ in $2^\omega$ such that, for all m, p, if $p=\mu q[\varepsilon ''(\overline{\gamma } m*\langle 0\rangle )<_\mathbb {Q} \delta '(q)\;\vee \; \delta ''(q) <_\mathbb {Q} \varepsilon '(\overline{\gamma } m*\langle 1 \rangle )]$, then $\gamma (m)=0\leftrightarrow \delta ''(p) <_\mathbb {Q} \varepsilon '(\overline{\gamma }m*\langle 1\rangle )$. One may prove, by induction on n, that, for each n, there exists p such that $\forall t\in 2^{<\omega }[\bigl (length(t)=n\;\wedge t\perp {\overline{\gamma }} n\bigr )\rightarrow \delta (p)\;\#_\mathbb {S}\;\varepsilon (t)]$. Assume that $\delta \;\#_\mathcal {R}\;\tau |\gamma$. Find n such that $\delta (n)\;\#_\mathbb {S} \;(\tau |\gamma )(n)=\varepsilon ({\overline{\gamma }}(n))$. Find $p>n$ such that $\forall t\in 2^{<\omega }[\bigl (length(t)=n \;\wedge \; t\perp {\overline{\gamma }} n\bigr )\rightarrow \delta (p) \;\#_\mathbb {S} \;\varepsilon (t)]$. Conclude that $\forall t \in 2^{<\omega }[length(t)=n \rightarrow \delta (p)\;\#_\mathbb {S}\; \varepsilon (t)]$ and, for each $\alpha$, $\delta \in \mathcal {H}_{\chi |\alpha }$. We thus see that, if $\delta \;\#_\mathcal {R}\;\tau |\gamma$, then $\forall \alpha [\delta \in \mathcal {H}_{\chi |\alpha }]$.

(iv) Assume: $\delta \in [0,1]^\omega$. Define $\gamma$ in $2^\omega$ such that, for all n, m, p, if $p=\mu q[\varepsilon ''(\overline{\gamma ^{\upharpoonright n}} m*\langle 0\rangle )<_\mathbb {Q} (\delta ^{\upharpoonright n})'(q)\;\vee \; (\delta ^{\upharpoonright n})''(q) <_\mathbb {Q} \varepsilon '(\overline{\gamma ^{\upharpoonright n}} m*\langle 1 \rangle )]$, then $\gamma ^{\upharpoonright n}(m)=0\leftrightarrow \delta ''(p) <_\mathbb {Q} \varepsilon '(\overline{(\gamma ^{\upharpoonright n})}m*\langle 1\rangle )$. Conclude, following the argument for (iii), that $\forall n[\delta ^{\upharpoonright n} \;\#_\mathcal {R}\;\tau |(\gamma ^{\upharpoonright n})\rightarrow \delta ^{\upharpoonright n}\in \mathcal {H}_{\alpha ^n}]$. $\square$

13.9 Real functions from [0, 1] to $\mathcal {R}$

$\varphi :[0,1]\rightarrow _\mathcal {R} \mathcal {R}$ if and only if³⁶

• (1) $\forall n\in E_\varphi [n'\in \mathbb {S}\;\wedge \;n'' \in \mathbb {S}]$, and

• (2) $\forall m\in E_\varphi \forall n\in E_\varphi [m'\sqsubseteq _\mathbb {S} n'\rightarrow m'' \sqsubseteq _\mathbb {S} n'']$, and

• (3) $\forall \alpha \in [0,1]\forall n\exists m\exists s[(\alpha (m), s)\in E_\varphi \;\wedge \; s''-_\mathbb {Q}s'<_\mathbb {Q}\frac{1}{2^n}]$.

$\mathcal {R}^{[0,1]}:=\{\varphi \mid \varphi :[0,1] \rightarrow _\mathcal {R} \mathcal {R}\}$.

Assume: $\varphi :[0,1]\rightarrow _\mathcal {R}\mathcal {R}$.

We define, for each $\alpha$ in [0, 1], for each $\beta$ in $\mathcal {R}$,

$\varphi :\alpha \mapsto _\mathcal {R}\beta$ if and only if $\forall n\exists m\exists p\in E_\varphi [\alpha (m)\sqsubseteq _\mathbb {S} p'\;\wedge \;p''\sqsubseteq _\mathbb {S}\beta (n)]$.

For each $\alpha$ in [0, 1], we let $\varphi ^{`\mathcal {R}}(\alpha )$ be the element $\beta$ of $\mathcal {R}$ such that, for each n, $\beta (n)=double_\mathbb {S}(s)$, where s is the least t such that $t\in \mathbb {S}$ and $t''-_\mathbb {Q}t'=_\mathbb {Q} \frac{1}{2^n}$ and

$\exists p\le t \exists r\le t\exists m\le t[\varphi (r)= p+1 \;\wedge \; \alpha (m)\sqsubseteq _\mathbb {S} p' \;\wedge \; p''\sqsubseteq _\mathbb {S} t]$.

Note: $\varphi :\alpha \mapsto _\mathcal {R} \varphi ^{`\mathcal {R}}(\alpha )$.

13.10 Game-theoretic terminology

$s:n\rightarrow k\leftrightarrow \bigl (length(s)= n\;\wedge \;\forall j<n[s(j) <k]\bigr )$.

$Seq(n,l):=\{s\mid s:n\rightarrow l\}$.

Note that $Seq(n,2)=\{s\in 2^{<\omega }\mid length(t)=n\}=2^{<\omega }\cap \omega ^n$.

$c \in _{I} s \leftrightarrow \forall i[2i < \textit{length}(c) \rightarrow c(2i) = s\bigl ({\overline{c}} (2i)\bigr )]$.³⁷

$c \in _{II} t \leftrightarrow \forall i[2i+1 < \textit{length}(c) \rightarrow c(2i+1) = t\bigl ({\overline{c}} (2i+1)\bigr )]$.³⁸

(The numbers s, t should be thought of as strategies for player I, II, respectively.)

$c \in _{I} \sigma \leftrightarrow \forall i[2i < \textit{length}(c) \rightarrow c(2i) = \sigma \bigl ({\overline{c}} (2i)\bigr )]$.

$c \in _{II} \tau \leftrightarrow \forall i[2i+1 < \textit{length}(c) \rightarrow c(2i+1) = \tau \bigl ({\overline{c}} (2i+1)\bigr )]$.

$\gamma \in _{I} \sigma \leftrightarrow \forall i[\gamma (2i) = \sigma \bigl ({\overline{\gamma }} (2i)\bigr )]$.

$\gamma \in _{II} \tau \leftrightarrow \forall i[\gamma (2i+1) = \tau \bigl ({\overline{\gamma }} (2i+1)\bigr )]$.

$\gamma \in _{I} s \leftrightarrow \forall i[{\overline{\gamma }}(2i)<length(s)\rightarrow \gamma (2i) = s\bigl ({\overline{\gamma }} (2i)\bigr )]$.

$\gamma \in _{II} t \leftrightarrow \forall i[{\overline{\gamma }}(2i+1)<length(t)\rightarrow \gamma (2i+1) = t\bigl ({\overline{\gamma }} (2i+1)\bigr )]$.

$\omega \times \omega :=\{s\mid \textit{length}(s) = 2\}$.

$2\times \omega :=\{s\mid \textit{length}(s) = 2 \;\wedge \; s(0)<2\}$.

$\omega \times 2 :=\{s\mid \textit{length}(s)=2 \;\wedge \; s(1) <2\}$.

For each $n>0$, $(\omega \times 2)^n:=\{s\mid \textit{length}(s) = 2n\;\wedge \forall i< n[s(2i+1) < 2]\}$ and

$(\omega \times 2)^n \times \omega :=\{s\mid \textit{length}(s) = 2n+1\;\wedge \;\forall i< n[s(2i+1) <2 ]\}$.

$(\omega \times 2)^{<\omega }:=\bigcup _n(\omega \times 2)^n$.

$(\omega \times 2)^\omega :=\{\delta \mid \forall n[\delta (2n+1) < 2]\}$.

$Halfbin:= (\omega \times 2)^{<\omega }\cup \bigl ((\omega \times 2)^{<\omega }\times \omega \bigr ) =\bigcup _n\{{\overline{\gamma }} n\mid \gamma \in (\omega \times 2)^\omega \}$.