Splittings and Robustness for the Heine-Borel Theorem

Abstract

The Heine-Borel theorem for uncountable coverings has recently emerged as an interesting and central principle in higher-order Reverse Mathematics and computability theory, formulated as follows: $\textsf {HBU}$ is the Heine-Borel theorem for uncountable coverings given as $\cup _{x\in [0,1]}(x-\varPsi (x), x+\varPsi (x))$ for arbitrary $\varPsi :[0,1]\rightarrow {\mathbb R}^{+}$, i.e. the original formulation going back to Cousin (1895) and Lindelöf (1903). In this paper, we show that $\textsf {HBU}$ is equivalent to its restriction to functions continuous almost everywhere, an elegant robustness result. We also obtain a nice splitting $\textsf {HBU} \leftrightarrow [\textsf {WHBU} ^{+}+\textsf {HBC} _{0}+\textsf {WKL} ]$ where $\textsf {WHBU} ^{+}$ is a strengthening of Vitali’s covering theorem and where $\textsf {HBC} _{0}$ is the Heine-Borel theorem for countable collections (and not sequences) of basic open intervals, as formulated by Borel himself in 1898.

Supported by the Deutsche Forschungsgemeinschaft via the DFG grant SA3418/1-1.

A Reverse Mathematics: Second- and Higher-Order

1.1 A.1 Reverse Mathematics

Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics initiated around 1975 by Friedman [5, 6] and developed extensively by Simpson [26]. The aim of RM is to identify the minimal axioms needed to prove theorems of ordinary, i.e. non-set theoretical, mathematics. We refer to [27] for a basic introduction to RM and to [25, 26] for an overview of RM. The details of Kohlenbach’s higher-order RM may be found in [10], including the base theory $\textsf {RCA} _{0}^{\omega }$. The latter is connected to ${\textsf {RCA}}_{0}$ by the $\textsf {ECF}$-translation as follows.

Remark A.1

(The -interpretation). The (rather) technical definition of $\textsf {ECF}$ may be found in [29, p. 138, §2.6,]. Intuitively, the $\textsf {ECF}$-interpretation $[A]_{\textsf {ECF} }$ of a formula $A\in \textsf {{L}}_{\omega }$ is just A with all variables of type two and higher replaced by type one variables ranging over so-called ‘associates’ or ‘RM-codes’; the latter are (countable) representations of continuous functionals. The $\textsf {ECF}$-interpretation connects $\textsf {RCA} _{0}^{\omega }$ and ${\textsf {RCA}}_{0}$ (see [10, Prop. 3.1,]) in that if $\textsf {RCA} _{0}^{\omega }$ proves A, then ${\textsf {RCA}}_{0}$ proves $[A]_{\textsf {ECF} }$, again ‘up to language’, as ${\textsf {RCA}}_{0}$ is formulated using sets, and $[A]_{\textsf {ECF} }$ is formulated using types, i.e. using type zero and one objects.

In light of the widespread use of codes in RM and the common practise of identifying codes with the objects being coded, it is no exaggeration to refer to $\textsf {ECF}$ as the canonical embedding of higher-order into second-order arithmetic.

We now introduce the usual notations for common mathematical notions.

Definition A.2

(Real numbers and related notions in )

• a. Natural numbers correspond to type zero objects, and we use ‘$n^{0}$’ and ‘$n\in {\mathbb N}$’ interchangeably. Rational numbers are defined as signed quotients of natural numbers, and ‘$q\in {\mathbb Q}$’ and ‘$<_{{\mathbb Q}}$’ have their usual meaning.

• b. Real numbers are coded by fast-converging Cauchy sequences $q_{(\cdot )}:{\mathbb N}\rightarrow {\mathbb Q}$, i.e. such that $(\forall n^{0}, i^{0})(|q_{n}-q_{n+i}|<_{{\mathbb Q}} \frac{1}{2^{n}})$. We use Kohlenbach’s ‘hat function’ from [10, p. 289,] to guarantee that every $q^{1}$ defines a real number.

• c. We write ‘$x\in {\mathbb R}$’ to express that $x^{1}:=(q^{1}_{(\cdot )})$ represents a real as in the previous item and write $[x](k):=q_{k}$ for the k-th approximation of x.

• d. Two reals x, y represented by $q_{(\cdot )}$ and $r_{(\cdot )}$ are equal, denoted $x=_{{\mathbb R}}y$, if $(\forall n^{0})(|q_{n}-r_{n}|\le {2^{-n+1}})$. Inequality ‘$<_{{\mathbb R}}$’ is defined similarly. We sometimes omit the subscript ‘${\mathbb R}$’ if it is clear from context.

• e. Functions $F:{\mathbb R}\rightarrow {\mathbb R}$ are represented by $\varPi ^{1\rightarrow 1}$ mapping equal reals to equal reals, i.e. extensionality as in $(\forall x , y\in {\mathbb R})(x=_{{\mathbb R}}y\rightarrow \varPi (x)=_{{\mathbb R}}\varPi (y))$.

• f. Binary sequences are denoted ‘$f,g\in C$’ or ‘$f, g\in 2^{{\mathbb N}}$’. Elements of Baire space are given by $f^{1}, g^{1}$, but also denoted ‘$f, g\in {\mathbb N}^{{\mathbb N}}$’.

Notation A.3

(Finite sequences). The type for ‘finite sequences of objects of type $\rho$’ is denoted $\rho ^{*}$, which we shall only use for $\rho =0,1$. Since the usual coding of pairs of numbers goes through in $\textsf {RCA} _{0}^{\omega }$, we shall not always distinguish between 0 and $0^{*}$. Similarly, we assume a fixed coding for finite sequences of type 1 and shall make use of the type ‘$1^{*}$’. In general, we do not always distinguish between ‘$s^{\rho }$’ and ‘$\langle s^{\rho }\rangle$’, where the former is ‘the object s of type $\rho$’, and the latter is ‘the sequence of type $\rho ^{*}$ with only element $s^{\rho }$’. The empty sequence for the type $\rho ^{*}$ is denoted by ‘$\langle \rangle _{\rho }$’, usually with the typing omitted. Furthermore, we denote by ‘$|s|=n$’ the length of the finite sequence $s^{\rho ^{*}}=\langle s_{0}^{\rho },s_{1}^{\rho },\dots ,s_{n-1}^{\rho }\rangle$, where $|\langle \rangle |=0$, i.e. the empty sequence has length zero. For sequences $s^{\rho ^{*}}, t^{\rho ^{*}}$, we denote by ‘$s*t$’ the concatenation of s and t, i.e. $(s*t)(i)=s(i)$ for $i<|s|$ and $(s*t)(j)=t(|s|-j)$ for $|s|\le j< |s|+|t|$. For a sequence $s^{\rho ^{*}}$, we define $\overline{s}N:=\langle s(0), s(1), \dots , s(N-1)\rangle$ for $N^{0}<|s|$. For a sequence $\alpha ^{0\rightarrow \rho }$, we also write $\overline{\alpha }N=\langle \alpha (0), \alpha (1),\dots , \alpha (N-1)\rangle$ for any $N^{0}$. Finally, $(\forall q^{\rho }\in Q^{\rho ^{*}})A(q)$ abbreviates $(\forall i^{0}<|Q|)A(Q(i))$, which is (equivalent to) quantifier-free if A is.

1.2 A.2 Further Systems

We define some standard higher-order systems that constitute the counterpart of e.g. $\Pi _{1}^{1}\text {-}{\textsf {CA}}_{0}$ and ${\textsf {{Z}}}_{2}$. First of all, the Suslin functional ${\textsf {S}}^{2}$ is defined in [10] as:

The system $\Pi _{1}^{1}\text {-}{\textsf {CA}}_{0}^{\omega }\equiv \textsf {RCA} _{0}^{\omega }+({\textsf {S}}^{2})$ proves the same $\Pi _{3}^{1}$-sentences as $\Pi _{1}^{1}\text {-}{\textsf {CA}}_{0}$ by [20, Theorem 2.2,]. By definition, the Suslin functional ${\textsf {S}}^{2}$ can decide whether a $\varSigma _{1}^{1}$-formula as in the left-hand side of $({\textsf {S}}^{2})$ is true or false. We similarly define the functional ${\textsf {S}}_{k}^{2}$ which decides the truth or falsity of $\varSigma _{k}^{1}$-formulas from $\textsf {{L}}_{2}$; we also define the system $\Pi _{k}^{1}\text {-}\textsf {{CA}}_{0}^{\omega }$ as $\textsf {RCA} _{0}^{\omega }+({\textsf {S}}_{k}^{2})$, where $({\textsf {S}}_{k}^{2})$ expresses that ${\textsf {S}}_{k}^{2}$ exists. We note that the operators $\nu _{n}$ from [2, p. 129,] are essentially ${\textsf {S}}_{n}^{2}$ strengthened to return a witness (if existant) to the $\varSigma _{n}^{1}$-formula at hand.

Secondly, second-order arithmetic ${\textsf {{Z}}}_{2}$ readily follows from $\cup _{k}\Pi _{k}^{1}\text {-}\textsf {{CA}}_{0}^{\omega }$, or from:

and we therefore define ${\textsf {{Z}}}_{2}^{\varOmega }\equiv \textsf {RCA} _{0}^{\omega }+(\exists ^{3})$ and ${\textsf {{Z}}}_{2}^\omega \equiv \cup _{k}\Pi _{k}^{1}\text {-}\textsf {{CA}}_{0}^{\omega }$, which are conservative over ${\textsf {{Z}}}_{2}$ by [9, Cor. 2.6,]. Despite this close connection, ${\textsf {{Z}}}_{2}^{\omega }$ and ${\textsf {{Z}}}_{2}^{\varOmega }$ can behave quite differently, as discussed in e.g. [13, §2.2,]. The functional from $(\exists ^{3})$ is also called ‘$\exists ^{3}$’, and we use the same convention for other functionals.