The abstract type of the real numbers

Abstract

In finite type arithmetic, the real numbers are represented by rapidly converging Cauchy sequences of rational numbers. Ulrich Kohlenbach introduced abstract types for certain structures such as metric spaces, normed spaces, Hilbert spaces, etc. With these types, the elements of the spaces are given directly, not through the mediation of a representation. However, these abstract spaces presuppose the real numbers. In this paper, we show how to set up an abstract type for the real numbers. The appropriateness of our construction works in tandem with the bounded functional interpretation.

Appendix

In this section, we give a very brief description of the main result of the bounded functional interpretation with the abstract type $\mathrm {R}$. It is far from a complete description, as we omit many things. The reader should consult the available literature for details. Here is a short guide to the literature. The bounded functional interpretation was introduced in [8], and [5] presents a compact, yet reasonably complete description of the interpretation for the classical case in the arithmetical setting. The introduction of abstract types within the bounded functional interpretation appeared for the first time in [3]. It was used in applications in [7], but only very recently a proper account of the subject has been written, cf. [4]. Kohlenbach’s book [13] does not cover the bounded functional interpretation but it is a good reference for many of the notions pertinent to this interpretation.

The bounded functional interpretation relies crucially on the notion of strong majorizability due to Marc Bezem in [1]. We define the relation of strong majorizability for the finite type structure with ground types $\mathrm {N}$ and $\mathrm {R}$. To each type $\sigma$, we associate an arithmetical type $\hat{\sigma }$ according to the following clauses: $\hat{\mathrm {N}} = \hat{\mathrm {R}} = \mathrm {N}$ and $\widehat{\sigma \rightarrow \tau } = \hat{\sigma } \rightarrow \hat{\tau }$ (an arithmetical type is a type built only from the ground type $\mathrm {N}$). The majorizability relation $\le ^{*}_\sigma$ infixes between a functional f of type $\sigma$ and a functional g of arithmetical type $\hat{\sigma }$. It is given by the following clauses: (1) $n\le ^{*}_\mathrm {N}m$ is $n\le m$; (2) $x \le ^{*}_\mathrm {R}n$ is $|x| \le n$; and (3) if f is of type $\sigma \rightarrow \tau$ and g is of type $\hat{\sigma } \rightarrow \hat{\tau }$,

$$\begin{aligned} f \le ^{*}_\sigma g \,\, \text{ is } \,\,\, \forall x^\sigma , y^{\hat{\sigma }}, z^{\hat{\sigma }} \, (x \le ^{*}_\sigma z \wedge y \le ^{*}_{\hat{\sigma }} z \rightarrow fx \le ^{*}_\tau gz \wedge gy \le ^{*}_{\hat{\tau }} gz). \end{aligned}$$

Given the quantificational structure of this definition, the bounded functional interpretation cannot be set up with these relations. We must instead work with ersatz relations $\unlhd _\sigma$, given by primitive symbols of the language, which are governed only by universal truths coming from the corresponding $\le ^{*}_\sigma$. Let me explain. The bounded formulas of our finite type language with the abstract type $\mathrm {R}$ are the elements of the smallest class of formulas that contains the atomic formulas (including the new atomic formulas of the form $t\unlhd _\sigma q$) and is closed under propositional conectives and bounded quantifications of the form $\forall x \unlhd _\sigma t \, (\ldots )$, where the variable x does not occur in the term t. We can only accept in our theory $\mathsf {T} _{\mathbb {R}}$ universal closures of bounded formulas. For instance, we could accept as axioms all the universal closures of bounded formulas that, after flattening, are true in the standard structure. Here, by the flattening of a formula, we mean the formula obtained by replacing the symbols $\unlhd _\sigma$ by $\le ^{*}_\sigma$. The standard structure is the majorizability structure whose ground types are the natural numbers (for the type $\mathrm {N}$) and the real numbers (for the type $\mathrm {R}$).

There are three principles that are important for the bounded functional interpretation. We formulate these characteristic principles with single variables but the principles also include the tuple case.

• Monotone bounded choice $\mathsf{mAC }^{\omega }_{\mathsf{bd },{\mathrm {R}}}$:

  $\quad \tilde{\forall }x \tilde{\exists }y A(x,y) \rightarrow \tilde{\exists }f \tilde{\forall }x \tilde{\exists }y \unlhd fx \, A (x,y)$

• Bounded collection $\mathsf{bC }^{\omega }_{\mathsf{bd} ,{\mathrm {R}}}$:

  $\quad \forall x \unlhd z \exists y A (x,y) \rightarrow \exists w \forall x \unlhd z \exists y\unlhd w \, A (x,y).$

• Majorizability principles $\mathsf{MAJ }^{\omega }_{\mathrm {R}}$:

  $\quad \forall x \exists w \, (x \unlhd w)$

In the above, A is always a bounded formula. The quantifications $\tilde{\forall }x \, (\ldots )$ abbreviate so-called monotone quantifications of the form $\tilde{\forall }x\, (x\unlhd x \rightarrow (\ldots ))$. Note that they require that x be of arithmetical type. The contrapositive of (C1a) is a bounded collection principle, and (C2) is a majorizability principle. It can be argued that the contrapositive of (C1b) is also a bounded collection principle.

The main result of the bounded functional interpretation is the following conservation result:

Theorem

Suppose that

$$\begin{aligned} \mathsf {T} _{\mathbb {R}}+ \mathsf{mAC }^{\omega }_{\mathsf{bd },{\mathrm {R}}}+ \mathsf{bC }^{\omega }_{\mathsf{bd} ,{\mathrm {R}}}+ \mathsf{MAJ }^{\omega }_{\mathrm {R}}\vdash \forall x^\sigma \exists y^\tau A(x,y), \end{aligned}$$

where A is a bounded formula (whose free variables are among x and y). Then there is a closed monotone arithmetical term t of type $\hat{\sigma } \rightarrow \hat{\tau }$ such that,

$$\begin{aligned} \mathsf {T} _{\mathbb {R}}\vdash \forall a^{\hat{\sigma }} \forall x \unlhd _\sigma a \exists y\unlhd _\tau ta \, A(x,y). \end{aligned}$$

As a consequence, as long as the flattenings of the axioms of $\mathsf {T} _{\mathbb {R}}$ are true in the standard (majorizabilty) structure, then so is the sentence

$$\begin{aligned} \forall a^{\hat{\sigma }} \forall x \le ^{*}_\sigma a \exists y\le ^{*}_\tau ta \, A^*(x,y), \end{aligned}$$

where $A^*$ is the flatening of A. We further remark that full induction can be proved in the theory $\mathsf {T} _{\mathbb {R}}+ \mathsf{mAC }^{\omega }_{\mathsf{bd },{\mathrm {R}}}+ \mathsf{bC }^{\omega }_{\mathsf{bd} ,{\mathrm {R}}}+ \mathsf{MAJ }^{\omega }_{\mathrm {R}}$ because of the presence of the recursors (for this, see [4]).

The above result is a consequence of the soundness theorem of the bounded functional interpretation. For the sake of clarification, let me discuss a technical issue. In dialecta based interpretations (like Kohlenbach’s monotone interpretation), there is always the thorny issue of the interpretation of the contraction axiom. In dialectica based interpretations, atomic formulas must have characteristic functions so that definitions by cases become possible. However, the proof of the soundness theorem of the bounded functional interpretation sidesteps this issue because it does not require definitions by cases.

As mentioned in the introduction, the monotone functional interpretation requires that all the primitive constants of the formal system be majorizable, i.e., that for every constant c of type $\sigma$ there is a closed arithmetical term t of type $\hat{\sigma }$ such that $c \unlhd _\sigma t$. The bounded functional interpretation must meet this requirement as well. By postulating $\mathrm {inv}\unlhd _{\mathrm {N}\rightarrow \mathrm {R}} \lambda n.1$, we can set up our theory of real numbers so that it does. The reader should notice that the sentence expressing the field axiom saying that every nonzero element has an inverse is not the closure of a bounded formula. The bounded functional interpretation would turn this into:

$$\begin{aligned} \forall n \exists m (\forall x\unlhd n \exists y\unlhd m (x\ne 0 \rightarrow xy=1)). \end{aligned}$$

The flattening of this sentence is clearly false, even in fields (it is already false with $n=1$). This is the reason why we cannot have this field axiom in our axiomatization for the ground type $\mathrm {R}$.