1 Introduction
In a nutshell, we study basic third-order statements about continuous functions on ‘unrepresented’ metric spaces, i.e., the latter come without second-order representation, working in Kohlenbach’s higher-order Reverse Mathematics, as introduced in [Reference Kohlenbach19] and Section 1.1.2. We establish that certain (very specific) such statements are classified in the second-order Big Five of Reverse Mathematics, while most variations/generalisations are not provable from the latter, and much stronger systems. Thus, we generalise the results in [Reference Normann and Sanders32] to metric spaces, but restrict ourselves to continuous functions.
We believe these results to be of broad interest as the logic (and even mathematics) community should be aware of the influence representations have on some of the most basic objects, like continuous functions on compact metric spaces, that feature in undergraduate curricula in mathematics and physics.
Moreover, our results also shed new light on Kohlenbach’s proof mining program: as stated in [Reference Kohlenbach20, Section 17.1] or [Reference Kohlenbach21, Section 1], the success of proof mining often crucially depends on avoiding the use of separability conditions. By the results in this paper, avoiding such conditions seems to be a highly non-trivial affair.
We provide some background and motivation for these results in Section 1.1, including a gentle introduction to higher-order Reverse Mathematics. We formulate necessary definitions and axioms in Section 1.2 and prove our main results in Section 2. Finally, some foundational implications of our results, including related to the coding practise of Reverse Mathematics, are discussed in Section 3.
1.1 Motivation and background
1.1.1 Introduction
In a nutshell, the topic of this paper is the study of compact metric spaces in higher-order arithmetic; this section provides ample motivation for this study, as well as a detailed overview of our results.
Now, we assume familiarity with the program Reverse Mathematics, abbreviated ‘RM’ in the below. An introduction to RM for the mathematician-in-the-street may be found in [Reference Stillwell44], while [Reference Dzhafarov and Mummert10, Reference Simpson42] are textbooks on RM. We shall work in Kohlenbach’s higher-order RM, introduced in Section 1.1.2. In Section 1.1.3, we provide some motivation for higher-order RM (and this paper), based on the following items.
-
(a) Simplicity: the coding of continuous functions and other higher-order objects in second-order RM is generally not needed in higher-order RM.
-
(b) Scope: discontinuous
$\mathbb {R}\rightarrow \mathbb {R}$
-functions have been studied by Euler, Abel, Riemann, Fourier, and Dirichlet, i.e., the former are definitely part and parcel of ordinary mathematics. Discontinuous functions are studied directly in higher-order RM; the second-order approach via codes has its problems. -
(c) Generality: do the results of RM, like the Big Five phenomenon, depend on the coding practise of RM? Higher-order RM provides a (much needed, in our opinion) negative answer in the case of continuous functions.
Having introduced and motivated higher-order RM in Sections 1.1.2 and 1.1.3, we discuss the results of this paper in some detail in Section 1.1.4. As we will see, the motivation for the study of compact metric spaces in this paper is provided by items (a) and (c) above. In particular, we investigate whether the representation of metric spaces in second-order RM has an influence on the logical properties of third-order theorems about compact metric spaces. The answer turns out to be positive, for all but one very specific choice of definitions.
1.1.2 Higher-order Reverse Mathematics
We provide a gentle introduction to Kohlenbach’s higher-order RM, including the base theory
$\textsf {RCA}_{0}^{\omega }$
. Our focus is on intuitive understanding rather than full technical detail.
First of all, the language of
$\textsf {RCA}_{0}^{\omega }$
includes all finite types. In particular, the collection of all finite types
$\mathbf {T}$
is defined by the two clauses:
$$\begin{align*}\text{(i)}\ 0\in \mathbf{T}\ \text{and (ii) If}\ \sigma, \tau\in \mathbf{T}\ \text{then}\ ( \sigma \rightarrow \tau) \in \mathbf{T}, \end{align*}$$
where
$0$
is the type of natural numbers, and
$\sigma \rightarrow \tau $
is the type of mappings from objects of type
$\sigma $
to objects of type
$\tau $
. The following table provides an overview of the most common objects, their types, and their order.
One often identifies elements of Cantor space
$2^{\mathbb {N}}$
and subsets of
$\mathbb {N}$
, as the former can be viewed as characteristic functions for the latter. Similarly, a subset
$X\subset \mathbb {N}^{\mathbb {N}}$
is given by the associated characteristic function 
. In this paper, we shall mostly restricts ourselves to third-order objects.
Secondly, a basic axiom of mathematics is that functions map equal inputs to equal outputs. The axiom of function extensionality guarantees this behaviour and is included in
${\textsf {RCA}}_{0}^{\omega }$
. As an example, we write
$f=_{1}g$
in case
$(\forall n\in \mathbb {N})(f(n)=g(n) )$
and say that ‘f and g are equal elements of Baire space’. Any third-order
$Y^{1\rightarrow 1}$
thus satisfies the following instance of the axiom of function extensionality:
$$\begin{align} (\forall f, g\in \mathbb{N}^{\mathbb{N}})(f=_{1}g\rightarrow Y(f)=_{1}Y(g)). \end{align} $$
Now, the real number field
$\mathbb {R}$
is central to analysis and other parts of mathematics. The real numbers are defined in
${\textsf {RCA}}_{0}^{\omega }$
in exactly the same way as in
${\textsf {RCA}}_{0}$
, namely as fast-converging Cauchy sequences. Hence, the formulas ‘
$x\in \mathbb {R}$
’, ‘
$x<_{\mathbb {R}} y$
’, and ‘
$x=_{\mathbb {R}}y$
’ have their usual well-known meaning. To define functions on the reals, we let an ‘
$\mathbb {R}\rightarrow \mathbb {R}$
-function’ be any
$Y^{1\rightarrow 1}$
that satisfies
$$\begin{align} (\forall x, y\in \mathbb{R})(x=_{\mathbb{R}}y\rightarrow Y(x)=_{\mathbb{R}}Y(y)), \end{align}$$
which is the axiom of function extensionality relative to the (defined) equality ‘
$=_{\mathbb {R}}$
’. We stress that all symbols pertaining to the real numbers in (
E
ℝ→ℝ
) have their usual second-order meaning.
Thirdly,
${\textsf {RCA}}_{0}$
is a system of ‘computable mathematics’ that includes comprehension for
$\Delta _{1}^{0}$
-formulas and induction for
$\Sigma _{1}^{0}$
-formulas. The former allows one to build algorithms, e.g., via primitive recursion, while the latter certifies their correctness. The base theory
${\textsf {RCA}}_{0}^{\omega }$
includes axioms that prove
$\Delta _{1}^{0}$
-comprehension and
$\Sigma _{1}^{0}$
-induction, as expected. Moreover,
${\textsf {RCA}}_{0}^{\omega }$
includes the defining axiom of the recursor constant
$\mathbf {R}_{0}$
, namely that for
$m\in \mathbb {N}$
and
$f\in \mathbb {N}^{\mathbb {N}}$
, we have
$$ \begin{align} \mathbf{R}_{0}(f, m, 0):= m\ \mathrm{ and }\ \mathbf{R}_{0}(f, m, n+1):= f(n, \mathbf{R}_{0}(f, m, n)), \end{align} $$
which defines primitive recursion with second-order functions. We hasten to add that higher-order parameters are allowed; as an example, we could use in (1.1) the function
$f\in \mathbb {N}^{\mathbb {N}}$
defined as
$f(n):= Y(q_{n})$
for any
$Y:\mathbb {R}\rightarrow \mathbb {N}$
and where
$(q_{n})_{n\in \mathbb {N}}$
lists all rational numbers. The previous example also illustrates—to our mind—the need for lambda calculus notation, where we would simply define
$f\in \mathbb {N}^{\mathbb { N}}$
as
$\lambda n^{0}.Y(q_{n})$
, underscoring that n is the (only) variable and Y a parameter. The system
${\textsf {RCA}}_{0}^{\omega }$
includes the defining axioms for
$\lambda $
-abstraction via combinators.
Fourth, second-order RM includes many results about codes for continuous functions, and we would like to ‘upgrade’ these results to third-order functions that satisfy the usual ‘epsilon-delta’ definition of continuity. To this end,
${\textsf {RCA}}_{0}^{\omega }$
includes the following fragment of the Axiom of Choice, provable in
${\textsf {ZF}}$
:
$$ \begin{align*} (\forall f\in \mathbb{N}^{\mathbb{N}})(\exists n^{0}\in \mathbb{N})A(f, n)\rightarrow (\exists Y:\mathbb{N}^{\mathbb{N}}\rightarrow \mathbb{N})(\forall f\in \mathbb{N}^{\mathbb{N}})A(f, Y(f)), \ \ \quad \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (\mathsf{QF}\text{-}\mathsf{AC}^{1,0})\end{align*} $$
for any quantifier-free formula A. Now, the following formula:
$$ \begin{align} \Phi\subset \mathbb{N}\ \textit{is a total code for a continuous}\ \mathbb{N}^{\mathbb{N}}\rightarrow \mathbb{N}^{\mathbb{N}}\text{-}\textit{function} \end{align} $$
has exactly the form as in the antecedent of
${\textsf {QF-AC}}^{0,1}$
. Applying the latter to (1.2), we readily obtain a function
$Z^{1\rightarrow 1}$
such that
$Z(f)$
equals the value of
$\Phi $
at any
$f\in \mathbb {N}^{\mathbb {N}}$
. A similar argument goes through for codes of continuous
$\mathbb {R}\rightarrow \mathbb {R}$
-functions.
Finally,
${\textsf {RCA}}_{0}^{\omega }$
and
${\textsf {RCA}}_{0}$
prove the same second-order sentences, a fact that is proved via the so-called ECF-translation (see [Reference Kohlenbach19, Section 2]). In a nutshell, the latter replaces third- and higher-order objects by second-order codes for continuous functions, a concept not alien to second-order RM. We will discuss the coding of continuous functions in more detail in Section 1.1.3.
1.1.3 The coding practise of Reverse Mathematics
We discuss the coding practise of RM, which will be seen to provide motivation for Kohlenbach’s higher-order RM.
First of all, second-order RM makes use of the rather frugal language of second-order arithmetic, which only includes variables for natural numbers
$n\in \mathbb {N}$
and sets of natural numbers
$X\subset \mathbb {N}$
. As a result, higher-order objects like functions on the reals and metric spaces, are unavailable and need to be ‘represented’ or ‘coded’ by second-order objects. This coding practise can complicate basic definitions: the reader need only compare the one-line ‘epsilon-delta’ definition of continuity to the second-order definition in [Reference Simpson42, II.6.1]; the latter takes the better half of a page. Similar complications arise for metric spaces, where the reader can compare Definition 1.3 to [Reference Simpson42, I.8.6]. Hence, a framework that includes third- and higher-order objects provides a simpler approach.
Secondly, discontinuous functions have been studied in mathematics long before the advent of set theory, by ‘big name’ mathematicians like Euler, Dirichlet, Riemann, Abel, and Fourier, as discussed in [Reference Sanders37, Section 5.2]. Hence, basic properties of discontinuous functions are part of ordinary mathematics and should be studied in RM. Higher-order RM provides a natural framework for the study of discontinuous functions, as explored in detail in [Reference Kohlenbach19, Reference Normann and Sanders32]. By contrast, the study of discontinuous functions via codes is problematicFootnote 1 , as discussed in detail in [Reference Sanders37, Section 6.2.2].
Thirdly, a central objective of mathematical logic is the classification of mathematical statements in hierarchies according to their logical strength. An example due to Simpson is the Gödel hierarchy from [Reference Simpson43]. The goal of RM, namely finding the minimal axioms that prove a theorem of ordinary mathematics, fits squarely into this objective. Ideally, the place of a given statement in the hierarchy at hand does not depend greatly on the representation used. In particular, RM seeks to analyse theorems of ordinary mathematics ‘as they stand’. The following quotes from [Reference Simpson42, pages 32 and 137] illustrate this claim.
The typical constructivist response to a nonconstructive mathematical theorem is to modify the theorem by adding hypotheses or “extra data”. In contrast, our approach in this book is to analyze the provability of mathematical theorems as they stand […]
[…] we seek to draw out the set existence assumptions which are implicit in the ordinary mathematical theorems as they stand.
Essentially the same claim may be found in [Reference Dzhafarov and Mummert10, Section 10.5.2] and in many parts of the RM literature. The main point is always the same: RM ideally studies mathematics ‘as is’ without any logical additions.
Since the coding of continuous functions is conspicuously absent from mainstream mathematics, it is then a natural question whether the aforementioned coding has an influence on the classification of theorems in RM. The following theorem implies that at least the Big Five phenomenon of RM does not really depend on the coding of continuous functions on various spaces.
Theorem 1.1. The system
${\textsf {RCA}}_{0}^{\omega }$
proves the following for
$\mathbb {X}=\mathbb {N}^{\mathbb {N}}$
or
$\mathbb {X}=\mathbb {R}$
.
Let
$\Phi $
be a code for a continuous
$\mathbb {X}\rightarrow \mathbb {X}$
-function. There is a third-order
$F:\mathbb {X}\rightarrow \mathbb {X}$
such that
$F(x)$
equals the value of
$\Phi $
at x for any
$x\in \mathbb {X}$
.
The system
${\textsf {RCA}}_{0}^{\omega }+{\textsf {WKL}}_{0}$
proves the following for
$\mathbb {X}=2^{\mathbb {N}}$
or
$\mathbb {X}=[0,1]$
.
Let the third-order function
$F:\mathbb {X}\rightarrow \mathbb {X}$
be continuous on
$\mathbb {X}$
. Then there is a code
$\Phi $
such that
$F(x)$
equals the value of
$\Phi $
at any
$x\in \mathbb {X}$
.
Proof See [Reference Kohlenbach18, Section 4] and [Reference Normann and Sanders32, Section 2] for proofs.
As a corollary, we observe that over
${\textsf {RCA}}_{0}^{\omega }$
, the second-order axiom
${\textsf {WKL}}_{0}$
is equivalent to third-order theorems like for any third-order
$F:[0,1]\rightarrow \mathbb {R}$
, continuity implies boundedness. One could argue that continuous functions are ‘really’ second-order, but the reader should keep the previous sentence in mind nonetheless.
1.1.4 Metric spaces and higher-order Reverse Mathematics
Having introduced and motivated higher-order RM in the previous two sections, we can now discuss and motivate the results in this paper in some detail.
First of all, an immediate corollary of Theorem 1.1 is that over
${\textsf {RCA}}_{0}^{\omega }$
, the second-order axiom
${\textsf {WKL}}_{0}$
is equivalent to many basic third-order theorems from real analysis about continuous functions. Motivated by this observation, Dag Normann and the author show in [Reference Normann and Sanders32, Reference Sanders40, Reference Sanders41] that many third-order theorems from real analysis about (possibly) discontinuous functions on the reals, are equivalent to the second-order Big Five, over
${\textsf {RCA}}_{0}^{\omega }$
. Moreover, slight variations/generalisations of the function class at hand yield third-order theorems that are not provable from the Big Five and the same for much stronger systems like
${\textsf {{Z}}}_{2}^{\omega }+{\textsf {QF-AC}}^{0,1}$
from Section 1.2. It is then a natural question whether a similar phenomenon can be found in other parts of mathematics and RM.
Secondly, in this paper, we study a different kind of generalisation than the one in [Reference Normann and Sanders32]: rather than going beyond the continuous functions, we study properties of the latter on compact metric spaces. Now, the study of the latter in second-order RM proceeds via codes: a complete separable metric space is represented via a countable and dense subset, as can be gleaned from [Reference Simpson42, II.5.1] or [Reference Brown5]. By contrast, we use the standard textbook definition of metric space as in Definition 1.3 without any additional data except that we are dealing with sets of reals. This study is not just spielerei as avoiding separability is, e.g., important in proof mining, as follows.
[…] it is crucial to exploit the fact that the proof to be analyzed does not use any separability assumption on the underlying spaces […]. [Reference Kohlenbach21, Section 1]
It will turn out that for [the aforementioned uniformity conditions] to hold we—in particular—must not use any separability assumptions on the spaces. [Reference Kohlenbach20, page 377]
Thirdly, in light of the previous two paragraphs, it is then a natural question whether basic properties of compact metric spaces without separability conditions are provable from second-order (comprehension) axioms or not. Theorem 2.2 provides a (rather) negative answer: well-known theorems due to Ascoli, Arzelà, Dini, Heine, and Pincherle, formulated for metric spaces, are not provable in
${\textsf {{Z}}}_{2}^{\omega }$
, a conservative extension of
${\textsf {{Z}}}_{2}$
introduced in Section 1.2. We only study metric spaces
$(M, d)$
where M is a subset of the reals or Baire space, i.e., the metric
$d:M^{2}\rightarrow \mathbb { R}$
is just a third-order mapping. By contrast, some (very specific) basic properties of metric spaces are provable from the Big Five and related systems by Theorem 2.3.
Fourth, the negative results in this paper are established using the uncountability of
$\mathbb {R}$
as formalised by the following principles (see Section 1.2 for details).
-
•
${\textsf {NIN}}_{[0,1]}$
: there is no injection from
$[0,1]$
to
$\mathbb {N}$
. -
•
${\textsf {NBI}}_{[0,1]}$
: there is no bijection from
$[0,1]$
to
$\mathbb {N}$
.
In particular, these principles are not provable in relatively strong systems, like
${\textsf {{Z}}}_{2}^{\omega }$
from Section 1.2. In Section 2.1, we identify a long and robust list of theorems that imply
${\textsf {NBI}}_{[0,1]}$
or
${\textsf {NIN}}_{[0,1]}$
. We have shown in [Reference Normann and Sanders31, Reference Normann and Sanders32, Reference Sam36] that many third-order theorems imply
${\textsf {NIN}}_{[0,1]}$
while we only know few theorems that only imply
${\textsf {NBI}}_{[0,1]}$
. As will become clear in Section 2.2, metric spaces provide (many) elegant examples of the latter. We also refine our results in Section 2.2, including connections to the RM of weak König’s lemma and the Jordan decomposition theorem.
In conclusion, we show that many basic (third-order) properties of continuous functions on metric spaces cannot be proved from second-order (comprehension) axioms when we omit the second-order representation of these spaces. A central principle is the uncountability of the reals as formalised by
${\textsf {NBI}}_{[0,1]}$
introduced above. These results carry foundational implications, as discussed in Section 3.
1.2 Preliminaries and definitions
We introduce some definitions, like the notion of open set or metric space in RM, and axioms that cannot be found in [Reference Kohlenbach19]. We emphasise that we only study metric spaces
$(M, d)$
where M is a subset of
$\mathbb {N}^{\mathbb {N}}$
or
$\mathbb {R}$
, modulo the coding of finite sequencesFootnote
2
of reals. Thus, everything can be formalised in the language of third-order arithmetic, i.e., we do not really go much beyond analysis on the reals.
Zeroth of all, we need to define the notion of (open) set. Now, open sets are represented in second-order RM by countable unions of basic open balls, namely as in [Reference Simpson42, II.5.6]. In light of [Reference Simpson42, II.7.1], (codes for) continuous functions provide an equivalent representation over
${\textsf {RCA}}_{0}$
. In particular, the latter second-order representation is exactly the following definition restricted to (codes for) continuous functions, as can be found in [Reference Simpson42, II.6.1].
Definition 1.2.
-
• A set
$U\subset \mathbb {R}$
(and its complement
$U^{c}$
) is given by
$h_{U}:\mathbb {R}\rightarrow [0,1]$
where we say ‘
$x\in U$
’ if and only if
$h_{U}(x)>0$
. -
• A set
$U\subset \mathbb {R}$
is open if
$y\in U$
implies
$(\exists N\in \mathbb {N})(\forall z\in B(y, \frac {1}{2^{N}})(z\in U )$
. A set is closed if the complement is open. -
• A set
$U\subset \mathbb {R}$
is finite if there is
$N\in \mathbb {N}$
such that for any finite sequence
$(x_{0}, \dots , x_{N})$
, there is
$i\leq N$
with
$x_{i}\not \in U$
. We sometimes write ‘
$|U|\leq~N$
’.
Now, codes for continuous functions denote third-order functions in
${\textsf {RCA}}_{0}^{\omega }$
by [Reference Normann and Sanders32, Section 2], i.e., Definition 1.2 thus includes the second-order definition of open set. To be absolutely clear, combining [Reference Normann and Sanders32, Theorem 2.2] and [Reference Simpson42, II.7.1],
${\textsf {RCA}}_{0}^{\omega }$
proves
$$\begin{align*}\mathrm{[}\textit{a second-order code U for an open set}\mathrm{]} \textit{ represents an open set as in Definition 1.2}. \end{align*}$$
Assuming Kleene’s quantifier
$(\exists ^{2})$
defined below, Definition 1.2 is equivalent to the existence of a characteristic function for U; the latter definition is used in, e.g., [Reference Normann and Sanders27, Reference Normann and Sanders33]. The interested reader can verify that over
${\textsf {RCA}}_{0}^{\omega }$
, a set U as in Definition 1.2 is open if and only if
$h_{U}$
is lower semi-continuous.
First of all, we shall study metric spaces
$(M, d)$
as in Definition 1.3, where M comes with its own equivalence relation ‘
$=_{M}$
’ and the metric d satisfies the axiom of extensionality on M as follows:
$$\begin{align*}(\forall x, y, v, w\in M)\big([x=_{M}y\wedge v=_{M}w]\rightarrow d(x, v)=_{\mathbb{R}}d(y, w)\big). \end{align*}$$
Similarly, we use
$F:M\rightarrow \mathbb {R}$
to denote functions from M to
$\mathbb {R}$
; the latter satisfy
$$\begin{align} (\forall x, y\in M)(x=_{M}y\rightarrow F(x)=_{\mathbb{R}}F(y)), \end{align} $$
i.e., the axiom of function extensionality relative to M.
Definition 1.3. A functional
$d: M^{2}\rightarrow \mathbb {R}$
is a metric on M if it satisfies the following properties for
$x, y, z\in M$
:
-
(a)
$d(x, y)=_{\mathbb {R}}0 \leftrightarrow x=_{M}y$
, -
(b)
$0\leq _{\mathbb {R}} d(x, y)=_{\mathbb {R}}d(y, x), $
-
(c)
$d(x, y)\leq _{\mathbb {R}} d(x, z)+ d(z, y)$
.
We use standard notation like
$B_{d}^{M}(x, r)$
to denote
$\{y\in M: d(x, y)<r\}$
.
To be absolutely clear, quantifying over M amounts to quantifying over
$\mathbb {N}^{\mathbb {N}}$
or
$\mathbb {R}$
, perhaps modulo coding, i.e., the previous definition can be made in third-order arithmetic for the intents and purposes of this paper. The definitions of ‘open set in a metric space’ and related constructs are now clear mutatis mutandis.
Secondly, the following definitions are now standard, where we note that the first item is called ‘Heine–Borel compact’ in, e.g., [Reference Brown3, Reference Brown5]. Moreover, coded complete separable metric spaces as in [Reference Simpson42, I.8.2] are only weakly complete over
${\textsf {RCA}}_{0}$
.
Definition 1.4 (Compactness and around).
For a metric space
$(M, d)$
, we say that
$:$
-
•
$(M, d)$
is countably-compact if for any
$(a_{n})_{n\in \mathbb {N}}$
in M and sequence of rationals
$(r_{n})_{n\in \mathbb {N}}$
such that we have
$M\subset \cup _{n\in \mathbb {N}}B^{M}_{d}(a_{n}, r_{n})$
, there is
$m\in \mathbb {N}$
such that
$M\subset \cup _{n\leq m}B^{M}_{d}(a_{n}, r_{n})$
. -
•
$(M, d)$
is strongly countably-compact if for any sequence
$(O_{n})_{n\in \mathbb {N}}$
of open sets in M such that
$M\subset \cup _{n\in \mathbb {N}}O_{n}$
, there is
$m\in \mathbb {N}$
such that
$M\subset \cup _{n\leq m}O_{n}$
. -
•
$(M, d)$
is compact in case for any
$\Psi :M\rightarrow \mathbb {R}^{+}$
, there are
$x_{0}, \dots , x_{k}\in M$
such that
$\cup _{i\leq k}B_{d}^{M}(x_{i}, \Psi (x_{i}))$
covers M. -
•
$(M, d)$
is sequentially compact if any sequence has a convergent sub-sequence. -
•
$(M, d)$
is limit point compact if any infinite set in M has a limit point. -
•
$(M, d)$
is complete in case every Cauchy
$^3$
sequence converges. -
•
$(M, d)$
is weakly complete if every effectively
Footnote
3
Cauchy sequence converges. -
•
$(M, d)$
is totally bounded if for all
$k\in \mathbb {N}$
, there are
$w_{0}, \dots , w_{m}\in \mathbb {N}$
such that
$\cup _{i\leq m}B_{d}^{M}(w_{i}, \frac {1}{2^{k}})$
covers M. -
•
$(M, d)$
is effectively totally bounded if there is a sequence of finite sequences
$(w_{n})_{n\in \mathbb {N}}$
in M such that for all
$k\in \mathbb {N}$
and
$x\in M$
, there is
$i<|w_{k}|$
such that
$x\in B(w(i), \frac {1}{2^{k}})$
. -
• a set
$C\subset M$
is sequentially closed if for any sequence
$(w_{n})_{n\in \mathbb {N}}$
in C converging to
$w\in M$
, we have
$w\in C$
. -
•
$(M, d)$
has the Cantor intersection property if any sequence of nonempty closed sets with
$M\supseteq C_{0}\supseteq \dots \supseteq C_{n}\supseteq C_{n+1}$
, has a nonempty intersection, -
•
$(M, d)$
has the sequential Cantor intersection property if the sets in the previous item are sequentially closed. -
•
$(M, d)$
is separable if there is a sequence
$(x_{n})_{n\in \mathbb {N}}$
in M such that
$(\forall x\in M, k\in \mathbb {N})(\exists n\in \mathbb {N})( d(x, x_{n})<\frac {1}{2^{k}})$
.
Thirdly, full second-order arithmetic
${\textsf {{Z}}}_{2}$
is the ‘upper limit’ of second-order RM. The systems
${\textsf {{Z}}}_{2}^{\omega }$
and
${\textsf {{Z}}}_{2}^{\Omega }$
are conservative extensions of
${\textsf {{Z}}}_{2}$
by [Reference Hunter15, Corollary 2.6]. The system
${\textsf {{Z}}}_{2}^{\Omega }$
is
${\textsf {RCA}}_{0}^{\omega }$
plus Kleene’s quantifier
$(\exists ^{3})$
(see, e.g., [Reference Hunter15, Reference Normann and Sanders32]), while
${\textsf {{Z}}}_{2}^{\omega }$
is
${\textsf {RCA}}_{0}^{\omega }$
plus
$({\textsf {S}}_{k}^{2})$
for every
$k\geq 1$
; the latter axiom states the existence of a functional
${\textsf {S}}_{k}^{2}$
deciding
$\Pi _{k}^{1}$
-formulas in Kleene normal form. The system
$\Pi _{1}^{1}\text {-}{\textsf {CA}}_{0}^{\omega }\equiv {\textsf {RCA}}_{0}^{\omega }+({\textsf {S}}_{1}^{2})$
is a
$\Pi _{3}^{1}$
-conservative extension of
$\Pi _{1}^{1}\text {-}{\textsf {CA}}_{0}$
[Reference Sakamoto and Yamazaki35], where
${\textsf {S}}_{1}^{2}$
is also called the Suslin functional. We also write
${\textsf {ACA}}_{0}^{\omega }$
for
${\textsf {RCA}}_{0}^{\omega }+(\exists ^{2})$
where the latter is as follows:
$$\begin{align} (\exists E:\mathbb{N}^{\mathbb{N}}\rightarrow \{0,1\})(\forall f \in \mathbb{N}^{\mathbb{N}})\big[(\exists n\in \mathbb{N})(f(n)=0) \leftrightarrow E(f)=0 \big]. \end{align} $$
Over
${\textsf {RCA}}_{0}^{\omega }$
,
$(\exists ^{2})$
is equivalent to the existence of Feferman’s
$\mu $
(see [Reference Kohlenbach19, Proposition 3.9]), defined as follows for all
$f\in \mathbb {N}^{\mathbb { N}}$
:
$$\begin{align*}\mu(f):= \begin{cases} n, & \text{if}\ n\ \text{is the least natural such that}\ f(n)=0,\\ 0, & \text{if}\ f(n)>0\ \text{for all}\ n\in \mathbb{N.} \end{cases} \end{align*}$$
Fourth, the uncountability of the reals, formulated as follows, is studied in [Reference Normann and Sanders31].
-
•
${\textsf {NIN}}_{[0,1]}$
: there is no
$Y:[0,1]\rightarrow \mathbb {N}$
that is injectiveFootnote
4
. -
•
${\textsf {NBI}}_{[0,1]}$
: there is no
$Y:[0,1]\rightarrow \mathbb {N}$
that is both injective and surjectiveFootnote
5
.
It is shown in [Reference Normann and Sanders30, Reference Normann and Sanders31] that
${\textsf {{Z}}}_{2}^{\omega }$
cannot prove
${\textsf {NBI}}_{[0,1]}$
and that
${\textsf {{Z}}}_{2}^{\omega }+{\textsf {QF-AC}}^{0,1}$
cannot prove
${\textsf {NIN}}_{[0,1]}$
, where the latter is countable choiceFootnote
6
for quantifier-free formulas. Moreover, many third-order theorems imply
${\textsf {NIN}}_{[0,1]}$
, as also established in [Reference Normann and Sanders31]. By contrast, that
$\mathbb {R}$
cannot be enumerated is formalised by Theorem 1.5.
Theorem 1.5. For any sequence of distinct real numbers
$(x_{n})_{n\in \mathbb {N}}$
and any interval
$[a,b]$
, there is
$y\in [a,b]$
such that y is different from
$x_{n}$
for all
$n\in \mathbb {N}$
.
The previous theorem is rather tame, especially compared to
${\textsf {NIN}}_{[0,1]}$
. Indeed, [Reference Gray13] includes an efficient computer program that computes the number y from Theorem 1.5 in terms of the other data; a proof of Theorem 1.5 in
${\textsf {RCA}}_{0}$
can be found in [Reference Simpson42, II.4.9], while a proof in Bishop’s Constructive Analysis is found in [Reference Bishop2, page 25].
Finally, the following remark discusses an interesting aspect of
$(\exists ^{2})$
and
${\textsf {NIN}}_{[0,1]}$
.
Remark 1.6 (On excluded middle).
Despite the grand stories told in mathematics and logic about Hilbert and the law of excluded middle, the ‘full’ use of the latter law in RM is almost somewhat of a novelty. To be more precise, the law of excluded middle as in
$(\exists ^{2})\vee \neg (\exists ^{2})$
is extremely useful, namely as follows: suppose we are proving
$T\rightarrow {\textsf {NIN}}_{[0,1]}$
over
${\textsf {RCA}}_{0}^{\omega }$
. Now, in case
$\neg (\exists ^{2})$
, all functions on
$\mathbb {R}$
(and
$\mathbb {N}^{\mathbb {N}}$
) are continuous by [Reference Kohlenbach19, Proposition 3.12]. Clearly, any continuous
$Y:[0,1]\rightarrow \mathbb {N}$
is not injective, i.e.,
${\textsf {NIN}}_{[0,1]}$
follows in the case that
$\neg (\exists ^{2})$
. Hence, what remains is to establish
$T\rightarrow {\textsf {NIN}}_{[0,1]}$
in case we have
$(\exists ^{2})$
. However, the latter axiom, e.g., implies
${\textsf {ACA}}_{0}$
(and sequential compactness) and can uniformly convert reals to their binary representations, which can simplify the remainder of the proof.
Here,
${\textsf {NIN}}_{[0,1]}$
is just one example and there are many more, all pointing to a more general phenomenon: while invoking
$(\exists ^{2})\vee \neg (\exists ^{2})$
may be non-constructive, it does lead to a short proof via case distinction: in case
$(\exists ^{2})$
, one has access to a stronger system while in case
$\neg (\exists ^{2})$
, the theorem at hand is a triviality (like for
${\textsf {NIN}}_{[0,1]}$
in the previous paragraph), or at least has a well-known second-order proof. We can also work over
${\textsf {RCA}}_{0}^{\omega }+{\textsf {WKL}}_{0}$
, noting that the latter establishes that continuous functions on
$[0,1]$
or
$2^{\mathbb {N}}$
have RM-codes (see [Reference Normann and Sanders32, Section 2] and [Reference Kohlenbach18, Section 4]).
2 Analysis on unrepresented metric spaces
We show that some (very specific) properties of continuous functions on compact metric spaces are classified in the (second-order) Big Five systems of Reverse Mathematics (Section 2.2), while most variations/generalisations are not provable from the latter, and much stronger systems (Section 2.1). The negative results are (mostly) established by deriving
${\textsf {NBI}}_{[0,1]}$
(Theorem 2.2), which is not provable in
${\textsf {{Z}}}_{2}^{\omega }$
. We also show that
${\textsf {NIN}}_{[0,1]}$
does not follow in most cases (Theorem 2.4).
2.1 Obtaining the uncountability of the reals
In this section, we show that basic properties of continuous functions on compact metric spaces, like Heine’s theorem in item (b), imply the uncountability of the reals as in
${\textsf {NBI}}_{[0,1]}$
. These basic properties are therefore not provable in
${\textsf {{Z}}}_{2}^{\omega }$
.
First of all, fragments of the induction axiom are sometimes used in an essential way in second-order RM (see, e.g., [Reference Neeman22]). The equivalence between induction and bounded comprehension is also well-known in second-order RM [Reference Simpson42, X.4.4]. We seem to need a little bit of the induction axiom as follows.
Principle 2.1
${\textsf {IND}}_{1}$
.
Let
$Y^{2}$
satisfy
$(\forall n \in \mathbb {N}) (\exists !f \in 2^{\mathbb {N}})[Y(n,f)=0]$
. Then
$ (\forall n\in \mathbb {N})(\exists w^{1^{*}})\big [ |w|=n\wedge (\forall i < n)(Y(i,w(i))=0)\big ]$
.
Note that
${\textsf {IND}}_{1}$
is a special case of the axiom of finite choice, and is valid in all models considered in [Reference Normann and Sanders23–Reference Normann and Sanders29, Reference Normann and Sanders31], i.e.,
${\textsf {{Z}}}_{2}^{\omega }+{\textsf {IND}}_{1}$
cannot prove
${\textsf {NBI}}_{[0,1]}$
. We have (first) used
${\textsf {IND}}_{1}$
in the RM of the Jordan decomposition theorem in [Reference Normann and Sanders30].
Secondly, the items in Theorem 2.2 are essentially those in [Reference Brown5, Theorem 4.1] or [Reference Simpson42, IV.2.2], but without codes. Equivalences of certain (coded) definitions of compactness are studied in second-order RM in, e.g., [Reference Brown3, Reference Brown4].
Theorem 2.2 (
${\textsf {RCA}}_{0}^{\omega }+{\textsf {IND}}_{1}$
).
The principle
${\textsf {NBI}}_{[0,1]}$
follows from any of the items (a)–(s) where
$(M, d)$
is a metric space with
$M\subset \mathbb {R}$
.
-
(a) For countably-compact
$(M, d)$
and sequentially continuous
$F:M\rightarrow \mathbb {R}$
, F is bounded on M. -
(b) Item (a) with ‘bounded’ replaced by ‘uniformly continuous’.
-
(c) Item (a) with ‘bounded’ replaced by ‘has a supremum’.
-
(d) Item (a) with ‘bounded’ replaced by ‘attains a maximum’.
-
(e) A countably-compact
$(M, d)$
has the sequential Cantor intersection property. -
(f) A countably-compact metric space
$(M, d)$
is separable.
The previous items still imply
${\textsf {NBI}}_{[0,1]}$
if we replace ‘countably-compact’ by ‘compact’ or ‘(weakly) complete and totally bounded’ or ‘strongly countably-compact’.
-
(h) For sequentially compact
$(M, d)$
, any continuous
$F:M\rightarrow \mathbb {R}$
is bounded. -
(i) Item (h) with ‘bounded’ replaced by ‘uniformly continuous’.
-
(j) Item (h) with ‘bounded’ replaced by ‘has a supremum’.
-
(k) Item (h) with ‘bounded’ replaced by ‘attains a maximum’.
-
(l) Items (h)–(k) assuming a modulus of continuity.
-
(m) Dini’s theorem ([Reference Berger and Schuster1, Reference Dini8, Reference Dini9]). Let
$(M, d)$
be sequentially compact and let
$F_{n}: (M\times \mathbb {N})\rightarrow \mathbb {R}$
be a monotone sequence of continuous functions converging to continuous
$F:M\rightarrow \mathbb {R}$
. Then the convergence is uniform. -
(n) On a sequentially compact metric space
$(M, d)$
, equicontinuity implies uniform equicontinuity. -
(o) (Pincherle [Reference Pincherle34, page 67]). For sequentially compact
$(M, d)$
and continuous
$F:M\rightarrow \mathbb {R}^{+}$
, we have
$(\exists k\in \mathbb {N})(\forall w\in M)(F(w)> \frac {1}{2^{k}} )$
. -
(p) (Ascoli-Arzelà, [Reference Simpson42, III.2]). For sequentially compact
$(M, d)$
, a uniformly bounded and equicontinuous sequence of functions on M has a uniformly convergent sub-sequence. -
(q) Any sequentially compact
$(M, d)$
is strongly countably-compact. -
(r) Any sequentially compact
$(M, d)$
is separable. -
(s) Any sequentially compact
$(M, d)$
has the seq. Cantor intersection property. -
(t) A sequentially compact metric space
$(M, d)$
is limit point compact.
Items (h)–(l) are provable in
${\textsf {{Z}}}_{2}^{\Omega }$
(via the textbook proof).
Proof First of all, by Remark 1.6, we may assume
$(\exists ^{2})$
as
${\textsf {NBI}}_{[0,1]}$
is trivial in case
$\neg (\exists ^{2})$
. Now suppose
$Y:[0,1]\rightarrow \mathbb {N}$
is a bijection, i.e., injective and surjective. Define M as the union of the new symbol
$\{0_{M}\}$
and the set
$N:=\{ w^{1^{*}}: (\forall i<|w|)(Y(w(i))=i)\}$
. We define ‘
$=_{M}$
’ as
$0_{M}=_{M}0_{M}$
,
$u\ne _{M} 0_{M}$
for
$u\in N$
, and
$w=_{M}v$
if
$w=_{1^{*}}v$
and
$w, v\in N$
. The metric
$d:M^{2}\rightarrow \mathbb {R}$
is defined as
$d(0_{M}, 0_{M})=_{\mathbb {R}}0$
,
$d(0_{M}, u)=d(u, 0_{M})=\frac {1}{2^{|u|}}$
for
$u\in N$
and
$d(w, v)=|\frac {1}{2^{|v|}}-\frac {1}{2^{|w|}}|$
for
$w, v\in N$
. Since Y is an injection, we have
$d(v, w)=_{\mathbb {R}}0 \leftrightarrow v=_{M}w$
. The other properties of a metric space from Definition 1.3 follow by definition (and the triangle equality of the absolute value on the reals).
Secondly, to show that
$(M, d)$
is countably-compact, fix a sequence
$(a_{n})_{n\in \mathbb {N}}$
in M and a sequence of rationals
$(r_{n})_{n\in \mathbb {N}}$
such that we have
$M\subset \cup _{n\in \mathbb { N}}B^{M}_{d}(a_{n}, r_{n})$
Suppose
$0_{M}\in B_{d}^{M}(a_{n_{0}}, r_{n_{0}})$
for
$a_{n_{0}}\ne _{M}0_{M}$
, i.e.,
$\frac {1}{2^{|a_{n_{0}}|}}=d(0_{M}, a_{n_{0}})<r_{n_{0}}$
. Then
$|\frac {1}{2^{|y|}}-\frac {1}{2^{|a_{n_{0}}|}}| =d(y, a_{n_{0}})<r_{n_{0}}$
holds for all
$y\in N$
such that
$|y|>|a_{n_{0}}|$
. Now use
${\textsf {IND}}_{1}$
to enumerate the (finitely many) reals
$z\in M$
with
$|z|<|a_{n_{0}}|$
. In this way, there exists a finite sub-covering of
$\cup _{n\in \mathbb {N}}B^{M}_{d}(a_{n}, r_{n})$
of at most
$|a_{n_{0}}|+1$
elements. The proof is analogous (and easier) in case
$a_{n_{0}}=_{M}0_{M}$
. Thus,
$(M, d)$
is a countably-compact metric space.
Thirdly, define the function
$F: M\rightarrow \mathbb {R}$
as follows:
$F(0_{M}):= 0$
and
$F(w):=|w|$
for any
$w\in N$
. Clearly, if the sequence
$(w_{n})_{n\in \mathbb {N}}$
in M converges to
$0_{M}$
, either it is eventually constant
$0_{M}$
or lists all reals in
$[0,1]$
. The latter case is impossible by Theorem 1.5. Hence, F is sequentially continuous at
$0_{M}$
, but not continuous at
$0_{M}$
. To show that F is (sequentially) continuous at
$w\ne 0_{M}$
, consider the formula
$|\frac {1}{2^{|w|}}-\frac {1}{2^{|v|}}|=d(v, w)<\frac {1}{2^{N}}$
; the latter is false for
$N\geq |w|+2$
and any
$v \ne _{M}0_{M}$
. Thus, the following formula is (vacuously) true:
$$\begin{align*}\textstyle (\forall k\in \mathbb{N})(\exists N\in \mathbb{N})(\forall v\in B_{d}^{M}(w, \frac{1}{2^{N}}))(|F(w)-F(v)|< \frac{1}{2^{k}}). \end{align*}$$
i.e., F is continuous at
$w\ne _{M}0_{M}$
, with a (kind of) modulus of continuity given. Applying item (a) (or item (c) and (d)), we obtain a contradiction as F is clearly unbounded on M. This contradiction yields
${\textsf {NBI}}_{[0,1]}$
and the same for item (b) as F is not (uniformly) continuous.
Fourth, to obtain
${\textsf {NBI}}_{[0,1]}$
from item (e), suppose again the former is false and
$Y:[0,1]\rightarrow \mathbb {R}$
and
$(M, d)$
are as above. Define
$C_{n}:= \{x\in N: |x|> n+1 \}$
and note that this set is non-empty (as Y is a surjection) but satisfies
$\cap _{n} C_{n}=\emptyset $
. Item (e) now yields a contradiction if we can show that
$C_{n}$
is sequentially closed. To the latter end, let
$(w_{k})_{k\in \mathbb {N}}$
be a sequence in
$C_{n}$
with limit
$w\in M$
. In case
$w=_{M}0_{M}$
, we make the same observation as in the third paragraph: either the sequence
$(w_{k})_{k\in \mathbb {N}}$
is eventually constant
$0_{M}$
or enumerates the reals in
$[0,1]$
. Both are impossible, i.e., this case does not occur. In case
$w\ne 0_{M}$
, we have
$$\begin{align*}\textstyle (\forall k\in \mathbb{N})(\exists N\in \mathbb{N})(\forall n\geq N)(|\frac{1}{2^{|w|}}-\frac{1}{2^{|w_{n}|}}| =d(w, w_{n})<\frac{1}{2^{k}}), \end{align*}$$
which is only possible if
$(w_{n})_{n\in \mathbb {N}}$
is eventually constant w. In this case of course,
$w\in C_{n}$
, i.e.,
$C_{n}$
is sequentially closed, and
$(e)\rightarrow {\textsf {NBI}}_{[0,1]}$
follows. Regarding item (f), suppose
$(M, d)$
is separable, i.e., there is a sequence
$(w_{n})_{n\in \mathbb {N}}$
such that
$$ \begin{align}\textstyle (\forall w\in M, k\in \mathbb{N})(\exists n\in \mathbb{N})( |\frac{1}{2^{|w|}}-\frac{1}{2^{|w_{n}|}}|=d(w, w_{n})<\frac{1}{2^{k}}). \end{align} $$
As in the above, for
$w\ne _{M} 0_{M}$
and
$k_{0}=|w|+2$
, the formula
$d(w, w_{n})<\frac {1}{2^{k_{0}}}$
is false for any
$n\in \mathbb {N}$
, i.e., we also obtain a contradiction in this case, yielding
${\textsf {NBI}}_{[0,1]}$
.
Fifth, for the sentences between items (f) and (h),
$(M, d)$
is also complete and (strongly countably) compact, which is proved in (exactly) the same way as in the second paragraph: any ball around
$0_{M}$
covers ‘most’ of M; to show that
$(M, d)$
is complete, let
$(w_{n})_{n\in \mathbb {N}}$
be a Cauchy sequence, i.e., we have
$$\begin{align*}\textstyle (\forall k\in \mathbb{N})(\exists N\in \mathbb{N})(\forall n, m\geq N)( d(w_{n}, w_{m})<\frac{1}{2^{k}}). \end{align*}$$
Then
$(w_{n})_{n\in \mathbb {N}}$
is either eventually constant or enumerates all reals in
$[0,1]$
. The latter is impossible by Theorem 1.5, i.e.,
$(w_{n})_{n\in \mathbb {N}}$
converges to some
$w\in M$
. Note that a continuous function is trivially sequentially continuous.
Sixth, to obtain
${\textsf {NBI}}_{[0,1]}$
from item (h) and higher, recall the set
$N:=\{ w^{1^{*}}: (\forall i<|w|)(Y(w(i))=i)\}$
and consider
$(N, d)$
, which is a metric space in the same way as for
$(M, d)$
. To show that
$(N, d)$
is sequentially compact, let
$(w_{n})_{n\in \mathbb {N}}$
be a sequence in N. In case
$(\forall n\in \mathbb {N})( |w_{n}|<m)$
for some
$m\in \mathbb {N}$
, then
$(w_{n})_{n\in \mathbb {N}}$
contains at most m different elements, as Y is an injection. The pigeon hole principle now implies that (at least) one
$w_{n_{0}}$
occurs infinitely often in
$(w_{n})_{n\in \mathbb {N}}$
, yielding an obviously convergent sub-sequence. In case
$(\forall m\in \mathbb {N})(\exists n\in \mathbb {N})( |w_{n}|\geq m)$
, the sequence
$(w_{n})_{n\in \mathbb {N}}$
enumerates the reals in
$[0,1]$
(as Y is a bijection), which is impossible by Theorem 1.5. Thus,
$(N, d)$
is a sequentially compact space; the function
$G:N\rightarrow \mathbb {R}$
defined as
$G(u)=|u|$
is continuous (in the same way as for F above) but not bounded. This contradiction establishes that item (h) implies
${\textsf {NBI}}_{[0,1]}$
, and the same for items (g)–(k). For item (l), the function
$H(x,k):= \frac {1}{2^{|x|+k+2}}$
is a modulus of continuity for G.
Seventh, for item (m), assume again
$\neg {\textsf {NBI}}_{[0,1]}$
and define
$G_{n}(w)$
as
$|w|$
in case
$|w|\leq n$
, and
$0$
otherwise. As for G above,
$G_{n}$
is continuous and
$\lim _{n\rightarrow \infty }G_{n}(w)=G(w)$
for
$x\in N$
. Since
$G_{n}\leq G_{n+1}$
on N, item (m) implies that the convergence is uniform, i.e., we have
$$ \begin{align}\textstyle (\forall k\in \mathbb{N})(\exists m\in \mathbb{N})(\forall w\in N)(\forall n\geq m)( |G_{n}(w)-G(w)|<\frac{1}{2^{k}}), \end{align} $$
which is clearly false. Indeed, take
$k=1$
and let
$m_{1}\in \mathbb {N}$
be as in (2.2). Since Y is surjective,
${\textsf {IND}}_{1}$
provides
$w_{1}\in N$
of length
$m_{1}+1$
, yielding
$|G(w_{1})-G_{m_{1}}(w_{1})|=|(m_{1}+1)-0|>\frac {1}{2}$
, contradicting (2.2) and thus
${\textsf {NBI}}_{[0,1]}$
follows from item (m). For item (n),
$(G_{n})_{n\in \mathbb {N}}$
is equicontinuous by the previous, but not uniformly equicontinuous, just like for item (m) using a variation of (2.2). For item (o), the function
$J(w):=\frac {1}{2^{|w|}} $
is continuous on N in the same way as for
$F, G$
. However, assuming
$\neg {\textsf {NBI}}_{[0,1]}$
, J becomes arbitrarily small on N, contradicting item (o). For item (p), define
$J_{n}(w)$
as
$J(w)$
if
$|w|\leq n$
, and
$1$
otherwise. Similar to the previous,
$J_{n}$
converges to J, but not uniformly, i.e., item (p) also implies
${\textsf {NBI}}_{[0,1]}$
.
For item (q), note that
$O_{n}:= \{w\in N: |w|=n\} $
is open as
$B_{d}^{M}(v, \frac {1}{2^{n+2}})\subset O_{n}$
in case
$v\in O_{n}$
. Then
$\cup _{n\in \mathbb {N}}O_{n}$
covers N, assuming N (and
$\neg {\textsf {NBI}}_{[0,1]}$
) as above. However, there clearly is no finite sub-covering.
Finally, for items (r) and (s), the above proof for items (e) and (f) goes through without modification. For item (t), note that N is an infinite set in
$(N, d)$
without limit point. The final sentence speaks for itself: one uses
$(\exists ^{3})$
and
$(\mu ^{2})$
to obtain a modulus of continuity. For
$\varepsilon =1$
, the latter yields an uncountable covering, which has finite sub-covering assuming
$(\exists ^{3})$
by [Reference Normann and Sanders28, Theorem 4.1]. This immediately yields an upper bound while the supremum and maximum are obtained using the usual interval-halving technique using
$(\exists ^{3})$
.
We could restrict item (q) to R2-open sets [Reference Normann and Sanders27, Reference Normann and Sanders33], where the latter are open sets such that
$x\in U$
implies
$B(x, h_{U}(x))\subset U$
with the notation of Definition 1.2.
2.2 Variations on a theme
Lest the reader believe that third-order metric spaces are somehow irredeemable, we show that certain (very specific) variations of the items in Theorem 2.2 are provable in rather weak systems, sometimes assuming countable choice as in
${\textsf {QF-AC}}^{0,1}$
(Theorems 2.3 and 2.4). We also show that certain items in Theorem 2.2 are just very hard to prove by deriving some of the new ‘Big’ systems from [Reference Normann and Sanders30, Reference Normann and Sanders31, Reference Sanders38, Reference Sanders40], namely the Jordan decomposition theorem and the uncountability of
$\mathbb {R}$
as in
${\textsf {NIN}}_{[0,1]}$
(Theorem 2.6).
First, we establish the following theorem, which suggests a strong need for open sets as in Definition 1.2 if we wish to prove basic properties of metric spaces in the base theory, potentially extended with the Big Five. The fourth item should be contrasted with item (e) in Theorem 2.2. Many variations of the below results are of course possible based on the associated second-order results.
Theorem 2.3 (
${\textsf {RCA}}_{0}^{\omega }$
).
-
(a) For strongly countably open
$(M, d)$
, a continuous
$F:M\rightarrow \mathbb {R}$
is bounded. -
(b) Dini’s theorem for strongly countably-compact
$(M, d)$
. -
(c) Pincherle’s theorem for strongly countably-compact
$(M, d)$
. -
(d) A metric space
$(M, d)$
with the Cantor intersection property, is strongly countably-compact. -
(e) The following are equivalent
$:$
-
(e.1) weak König’s lemma
${\textsf {WKL}}_{0}$
, -
(e.2) for any weakly complete and effectively totally bounded metric space
$(M, d)$
with
$M\subset [0,1]$
, a continuous
$F:M\rightarrow \mathbb {R}$
is bounded above, -
(e.3) the previous item for sequentially continuous functions.
-
-
(f) The following are equivalent.
-
(f.1) arithmetical comprehension
${\textsf {ACA}}_{0}$
, -
(f.2) any weakly complete and effectively totally bounded metric space
$(M, d)$
with
$M\subset [0,1]$
, is sequentially compact.
-
Proof For the first item, since F is continuous, the set
$E_{n}:= \{ x\in M: |F(x)|>n \}$
is open and exists in the sense of Definition 1.2. Since
$\cup _{n\in \mathbb {N}}E_{n}$
covers
$(M, d)$
, there is a finite sub-covering
$\cup _{n\leq n_{0}}E_{n}$
for some
$n_{0}\in \mathbb {N}$
, implying
$|F(x)|\leq n_{0}+1$
for all
$x\in M$
, i.e., F is bounded as required.
For the second item, let
$F, F_{n}$
be as in Dini’s theorem and define
$G_{n}(w):=F(w)-F_{n}(w)$
. Now fix
$k\in \mathbb {N}$
and define
$E_{n}:= \{w\in M: G_{n}(w)<\frac {1}{2^{k}} \}$
. The latter yields a countable open covering and one obtains uniform convergence from any finite sub-covering. For the third item, fix
$F:M\rightarrow \mathbb {R}^{+}$
and define
$E_{n}:=\{w\in M: F(w)>\frac {1}{2^{n}} \} $
. The proof proceeds as for the previous items.
For the fourth item, this amounts to a manipulation of definitions. For the fifth item, that (e.2) and (e.3) imply
${\textsf {WKL}}_{0}$
is immediate by [Reference Normann and Sanders32, Theorem 2.8] for
$M=[0,1]$
and [Reference Kohlenbach19, Proposition 3.6]. For the downward implication, fix
$F:M\rightarrow \mathbb {R}$
for
$M\subset [0,1]$
as in item (e.2). In case
$\neg (\exists ^{2})$
, all functions on
$\mathbb {R}$
are continuous by [Reference Kohlenbach19, Proposition 3.12]. By [Reference Normann and Sanders32, Theorem 2.8], all (continuous)
$[0,1]\rightarrow \mathbb {R}$
-functions are bounded. Since we may (also) view F as a (continuous) function from reals to reals, F is bounded on
$[0,1]$
and hence M, i.e., this case is finished.
In case
$(\exists ^{2})$
, we follow the well-known proof to show that
$(M, d)$
is sequentially compact. Indeed, for a sequence
$(x_{n})_{n\in \mathbb {N}}$
in M, define a sub-sequence as follows: M can be covered by a finite number of balls with radius
$1/2^{k}$
with
$k=1$
. Find a ball with infinitely many elements of
$(x_{n})_{n\in \mathbb {N}}$
inside (which can be done explicitly using
$(\exists ^{2})$
) and choose
$x_{n_{0}}$
in this ball to define
$y_{0}:=x_{n_{0}}$
. Now repeat the previous steps for
$k>1$
and note that the resulting sequence is effectively Cauchy and hence convergent (by the assumptions on M). Hence,
$(M, d)$
is sequentially compact and suppose
$F:M\rightarrow \mathbb {R}$
is unbounded, i.e.,
$(\forall n\in \mathbb {N})\underline {(\exists x\in M)}(F(x)>n)$
. It is now important to note that the underlined quantifier can be replaced by a quantifier over
$\mathbb {N}$
using the sequence
$(w_{n})_{n\in \mathbb {N}}$
provided by M being effectively totally bounded. Applying
${\textsf {QF-AC}}^{0, 0}$
, included in
${\textsf {RCA}}_{0}^{\omega }$
, there is a sequence
$(x_{n})_{n\in \mathbb {N}}$
such that
$|F(x_{n})|>n$
. This sequence has a convergent sub-sequence, say with limit y, and F is not continuous at y, a contradiction. Thus, F is bounded for both disjuncts of
$(\exists ^{2})\vee \neg (\exists ^{2})$
. The equivalence involving
${\textsf {ACA}}_{0}$
has a similar proof.
The final part of the proof seems to crucially depend on effective totally boundedness. Indeed, by the first part of Theorem 2.2, item (e.3) of Theorem 2.3 with ‘effectively’ omitted, implies
${\textsf {NBI}}_{[0,1]}$
. In other words, the equivalences in Theorem 2.3 do not seem robust.
Secondly, we show that certain items from Theorem 2.2 fit nicely with RM, assuming an extended base theory. Other items turn out to be connected to the ‘new’ Big systems studied in [Reference Normann and Sanders30, Reference Sanders38, Reference Sanders39].
We now show that certain items from Theorem 2.2 are provable assuming countable choice as in
${\textsf {QF-AC}}^{0,1}$
. Thus, these items do not imply
${\textsf {NIN}}_{[0,1]}$
as the latter is not provable in
${\textsf {{Z}}}_{2}^{\omega }+{\textsf {QF-AC}}^{0,1}$
. The third item should be contrasted with [Reference Simpson42, III.2]. Many results in RM do not go through in the absence of
${\textsf {QF-AC}}^{0,1}$
, as studied at length in [Reference Normann and Sanders27, Reference Normann and Sanders28].
Theorem 2.4 (
${\textsf {RCA}}_{0}^{\omega }+{\textsf {QF-AC}}^{0,1}$
).
The following are provable for
$(M, d)$
any metric space with
$M\subset \mathbb {R}$
.
-
• Items (h), (g), (m), (n), (o), (q), (s), and (t) from Theorem 2.2.
-
• The following are equivalent
$:$
-
– weak König’s lemma
${\textsf {WKL}}_{0}$
, -
– the unit interval is strongly countably-compact.
-
-
• The following are equivalent
$:$
-
– arithmetical comprehension
${\textsf {ACA}}_{0}$
, -
– a weakly complete and effectively totally bounded
$(M, d)$
with
$M\subset [0,1]$
is limit point compact.
-
Proof First of all, we prove item (h) from Theorem 2.2 in
${\textsf {RCA}}_{0}^{\omega }+{\textsf {QF-AC}}^{0,1}$
. To this end, suppose the continuous function
$F:M\rightarrow \mathbb {R}$
is unbounded, i.e.,
$(\forall n\in \mathbb {N})(\exists w\in M)(|F(w)|>n)$
. Applying
${\textsf {QF-AC}}^{0, 1}$
, there is a sequence
$(x_{n})_{n\in \mathbb {N}}$
such that
$|F(w_{n})|>n$
. Since
$(M, d)$
is assumed to be sequentially complete, let
$(y_{n})_{n\in \mathbb {N}}$
be a convergent sub-sequence with limit
$y\in M$
. Clearly, F cannot be continuous at
$y\in M$
, a contradiction, which yields item (h). Item (g) is proved in the same way: suppose F is not uniformly continuous and apply
${\textsf {QF-AC}}^{0,1}$
to the latter statement to obtain a sequence. Then F is not continuous at the limit of the convergent sub-sequence. Items (m)–(o) are proved in the same way. To prove item (q), let
$(O_{n})_{n\in \mathbb {N}}$
be a countable open covering of M with
$(\forall n\in \mathbb {N})(\exists x\in M)( x\not \in \cup _{m\leq n}O_{m} )$
. Apply
${\textsf {QF-AC}}^{0,1}$
to obtain a sequence
$(x_{n})_{n\in \mathbb {N}}$
, which has a convergent sub-sequence
$(y_{n})_{n\in \mathbb {N}}$
by assumption, say with limit
$y\in M$
. Then
$y\in O_{n_{0}}$
for some
$n_{0}\in \mathbb {N}$
, which implies that
$y_{n}$
is also eventually in
$O_{n_{0}}$
, a contradiction. To prove item (s), let
$(C_{n})_{n\in \mathbb {N}}$
be as in the sequential Cantor intersection property and apply
${\textsf {QF-AC}}^{0,1}$
to
$(\forall n\in \mathbb {N})(\exists x\in M)( x\in C_{n})$
. The convergent sub-sequence has a limit
$y\in \cap _{n\in \mathbb {N}} C_{n}$
. To prove item (t), let X be an infinite set, i.e.,
$(\forall N\in \mathbb {N})(\exists w^{1^{*}})(\forall i<|w|)(|w|=N\wedge w(i)\in X )$
. Now apply
${\textsf {QF-AC}}^{0,1}$
to obtain a sequence
$(w_{n})_{n\in \mathbb {N}}$
in X. Since
$(M, d)$
is sequentially closed, the latter sequence has a convergent sub-sequence, the limit of which is a limit point of X.
Secondly, the equivalence in the second item is proved in [Reference Normann and Sanders27, Theorem 4.1]. For the third item, the upwards implication is immediate for
$M=[0,1]$
. For the downwards implication, assume
$(M, d)$
as in the final sub-item. Theorem 2.3 implies that
$(M, d)$
is sequentially compact. As in the previous paragraph, an infinite set in M now has a limit point.
A similar proof should go through for many of the other items in Theorem 2.2 and for
${\textsf {QF-AC}}^{0,1}$
replaced by
${\textsf {NCC}}$
from [Reference Normann and Sanders29]; the latter is provable in
${\textsf {{Z}}}_{2}^{\Omega }$
while the former is not provable in
${\textsf {ZF}}$
.
Secondly, the Jordan decomposition theorem is studied in [Reference Normann and Sanders30, Reference Sanders40] where various versions are shown to be equivalent to the enumeration principle for countable sets. Many equivalences exist for the following principle, elevating it to a new ‘Big’ system, as shown in [Reference Normann and Sanders30].
Principle 2.5 (
${\textsf {cocode}}_{0}$
).
Let
$A\subset [0,1]$
and
$Y:[0,1]\rightarrow \mathbb {N}$
be such that Y is injective on A. Then there is a sequence of reals
$(x_{n})_{n\in \mathbb {N}}$
that includes A.
This principle is ‘explosive’ in that
${\textsf {ACA}}_{0}^{\omega }+{\textsf {cocode}}_{0}$
proves
${\textsf {ATR}}_{0}$
and
$\Pi _{1}^{1}\text {-}{\textsf {CA}}_{0}^{\omega }+{\textsf {cocode}}_{0}$
proves
$\Pi _{2}^{1}\text {-}\textsf {{CA}}_{0}$
(see [Reference Normann and Sanders30, Section 4]). As it turns out, the separability of metric spaces is similarly explosive.
Theorem 2.6 (
${\textsf {ACA}}_{0}^{\omega }$
).
Proof For the first item, let
$Y:[0,1]\rightarrow \mathbb {N}$
be injective on
$A\subset [0,1]$
; without loss of generality, we may assume
$0\in A$
. Now define
$d(x, y):= |\frac {1}{2^{Y(x)}}-\frac {1}{2^{Y(y)}}|$
,
$d(x,0)=d(0, x):=\frac {1}{2^{Y(x)}}$
for
$x, y\ne 0$
and
$d(0, 0):=0$
. The metric space
$(A, d)$
is countably-compact as
$0\in B_{d}^{A}(x, r)$
implies
$y\in B_{d}^{A}(x, r)$
for
$y\in A$
with only finitely many exceptions (as Y is injective on A). Similarly,
$(A, d)$
is sequentially compact: in case a sequence
$(z_{n})_{n\in \mathbb {N}}$
in A has at most finitely many distinct elements, there is an obvious convergent/constant sub-sequence. Otherwise,
$(z_{n})_{n\in \mathbb {N}}$
has a sub-sequence
$(y_{n})_{n\in \mathbb {N}}$
such that
$Y(y_{n})$
becomes arbitrary large with n increasing; this sub-sequence is readily seen to converge to
$0$
.
Now let
$(x_{n})_{n\in \mathbb {N}}$
be the sequence provided by item (f) or (r) of Theorem 2.2, implying
$(\forall x\in A)(\exists n\in \mathbb {N})( d(x, x_{n})<\frac {1}{2^{Y(x)+1}})$
by taking
${k=Y(x)+1}$
. The latter formula implies
$$ \begin{align}\textstyle (\forall x\in A)(\exists n\in \mathbb{N})(x\ne_{\mathbb{R}} 0\rightarrow |\frac{1}{2^{Y(x)}}-\frac{1}{2^{Y(x_{n})}}|<_{\mathbb{R}}\frac{1}{2^{Y(x)+1}}) \end{align} $$
by definition. Note that
$x_{n}$
from (2.3) cannot be
$0$
by the definition of the metric d. Clearly,
$|\frac {1}{2^{Y(x)}}-\frac {1}{2^{Y(x_{n})}}|<\frac {1}{2^{Y(x)+1}}$
is only possible if
$Y(x)=Y(x_{n})$
, implying
$x=_{\mathbb {R}}x_{n}$
. Hence, we have shown that
$(x_{n})_{n\in \mathbb {N}}$
lists all reals in
$A\setminus \{0\}$
. The same proof now yields the second item for
$A=[0,1]$
as Theorem 1.5 implies the reals cannot be enumerated.
In conclusion, the coding of metric spaces does distort the logical properties of basic properties of continuous functions on metric spaces by Theorem 2.2. This is established by deriving
${\textsf {NBI}}_{[0,1]}$
while noting that
${\textsf {NIN}}_{[0,1]}$
generally does not follow by Theorem 2.4. The latter also shows that in an enriched base theory, one can obtain ‘rather vanilla’ RM. By contrast, other properties of metric spaces imply new ‘Big’ systems, as is clear from Theorem 2.6.
3 Foundational musings
3.1 Thoughts on coding
The results in this paper have implications for the coding of higher-order objects in second-order RM, as discussed in this section.
First of all, our results shed new light on the following problem from [Reference Friedman and Simpson11, page 135].
PROBLEM. […] Show that Simpson’s neighborhood condition coding of partial continuous functions between complete separable metric spaces is “optimal”.
A coding is called optimal in [Reference Friedman and Simpson11] in case
${\textsf {RCA}}_{0}$
can prove ‘as much as possible’, i.e., as many as possible of the basic properties of the coding can be established in
${\textsf {RCA}}_{0}$
. Theorem 2.2 show that without separability, basic properties of continuous functions on compact metric spaces are no longer provable from second-order (comprehension) axioms. Thus, separability is an essential ingredient if one wishes to study these matters using second-order arithmetic/axioms.
Secondly, second-order (comprehension) axioms can establish many (third-order) theorems about continuous and discontinuous functions on the reals (see [Reference Normann and Sanders32, Reference Sanders40]), assuming
${\textsf {RCA}}_{0}^{\omega }$
. Hence, large parts of (third-order) real analysis can be developed using second-order comprehension axioms in a weak third-order background theory, namely
${\textsf {RCA}}_{0}^{\omega }$
, using little-to-no-coding. The same does not hold for continuous functions on compact metric spaces by the above results. In particular, Theorem 2.3 suggests we have to choose a very specific representation, namely ‘weakly complete and effectively totally bounded’ to obtain third-order statements that are classified in the Big Five. Indeed, Theorem 2.2 implies that many (most?) other variations are not provable from second-order (comprehension) axioms.
In conclusion, our results show that separability is an essential ingredient if one wishes to study these matters using second-order arithmetic/axioms. However, our results also show that this is a very specific choice that is ‘non-standard’ in the sense that many variations cannot be established using second-order arithmetic/axioms.
3.2 Set theory and ordinary mathematics
In this section, we explore a theme introduced in [Reference Sanders39]. Intuitively speaking, we collect evidence for a parallel between our results and some central results in set theory. Formulated slightly differently, one could say that interesting phenomena in set theory have ‘miniature versions’ to be found in third-order arithmetic, or that the seeds for interesting phenomena in set theory can already be found in third-order arithmetic.
First of all, the cardinality of
$\mathbb {R}$
is mercurial in nature: the famous work of Gödel [Reference Gödel12] and Cohen [Reference Cohen6, Reference Cohen7] shows that the Continuum Hypothesis cannot be proved or disproved in
${\textsf {ZFC}}$
, i.e., Zermelo-Fraenkel set theory with
${\textsf {AC}}$
, the usual foundations of mathematics. In particular, the exact cardinality of
$\mathbb { R}$
cannot be established in
${\textsf {ZFC}}$
. A parallel observation in higher-order RM is that
${\textsf {{Z}}}_{2}^{\omega }+{\textsf {QF-AC}}^{0,1}$
cannot prove that
$\mathbb {R}$
is uncountable in the sense of there being no no injection from
$\mathbb { R}$
to
$\mathbb {N}$
(see [Reference Normann and Sanders31] for details). In a conclusion, the cardinality of
$\mathbb {R}$
has a particularly mercurial nature, in both set theory and higher-order arithmetic.
Secondly, many standard results in mainstream mathematics are not provable in
${\textsf {ZF}}$
, i.e.,
${\textsf {ZFC}}$
with
${\textsf {AC}}$
removed, as explored in great detail [Reference Herrlich14]. The absence of
${\textsf {AC}}$
is even said to lead to disasters in topology and analysis (see [Reference Keremedis17]). A parallel phenomenon was observed in [Reference Normann and Sanders27, Reference Normann and Sanders28], namely that certain rather basic equivalences go through over
${\textsf {RCA}}_{0}^{\omega }+{\textsf {QF-AC}}^{0,1}$
, but not over
${\textsf {{Z}}}_{2}^{\omega }$
.
Examples include the equivalence between compactness results and local–global principles, which are intimately related according to Tao [Reference Tao and Gowers45]. In this light, it is fair to say that disasters happen in both set theory and higher-order arithmetic in the absence of
${\textsf {AC}}$
. It should be noted that
${\textsf {QF-AC}}^{0,1}$
(not provable in
${\textsf {ZF}}$
) can be replaced by
${\textsf {NCC}}$
from [Reference Normann and Sanders29] (provable in
${\textsf {{Z}}}_{2}^{\Omega }$
) in the aforementioned.
Thirdly, we discuss the essential role of
${\textsf {AC}}$
in measure and integration theory, which leads to rather concrete parallel observations in higher-order arithmetic. Indeed, the full pigeonhole principle for measure spaces is not provable in
${\textsf {ZF}}$
, which immediately follows from, e.g., [Reference Herrlich14, Diagram 3.4]. A parallel phenomenon in higher-order arithmetic (see [Reference Sanders39]) is that even the restriction to closed sets, namely
${\textsf {PHP}}_{[0,1]}$
cannot be proved in
${\textsf {{Z}}}_{2}^{\omega }+{\textsf {QF-AC}}^{0,1}$
(but
${\textsf {{Z}}}_{2}^{\Omega }$
suffices).
A more ‘down to earth’ observation pertains to the intuitions underlying the Riemann and Lesbesgue integral. Intuitively, the integral of a non-negative function represents the area under the graph; thus, if the integral is zero, then this function must be zero for ‘most’ reals. Now,
${\textsf {AC}}$
is needed to establish this intuition for the Lebesgue integral [Reference Kanovei and Katz16]. Similarly, [Reference Sanders39, Theorem 3.8] establishes the parallel observation that this intuition for the Riemann integral cannot be proved in
${\textsf {{Z}}}_{2}^{\omega }+{\textsf {QF-AC}}^{0,1}$
(but
${\textsf {{Z}}}_{2}^{\Omega }$
suffices as usual).
Fourth, the pointwise equivalence between sequential and ‘epsilon-delta’ continuity cannot be proved in
${\textsf {ZF}}$
while
${\textsf {RCA}}_{0}^{\omega }+{\textsf {QF-AC}}^{0,1}$
suffices for functions on Baire space (see [Reference Kohlenbach19]). A parallel observation is provided by (the proof of) Theorem 2.2, namely that the following statement is not provable in
${\textsf {{Z}}}_{2}^{\omega }$
:
for countably-compact
$(M, d)$
and sequentially continuous
$F:M\rightarrow \mathbb {R}$
, F is continuous on M.
Thus, the global equivalence between sequential and ‘epsilon-delta’ continuity on metric spaces cannot be proved in
${\textsf {{Z}}}_{2}^{\omega }$
. In other words, the exact relation between sequential and ‘epsilon-delta’ continuity is hard to pin down, both in set theory and third-order arithmetic.
Acknowledgments
The initial ideas for this paper were developed in my 2022 Habilitation thesis at TU Darmstadt [Reference Sam36] under the guidance of Ulrich Kohlenbach.
Funding
This research was supported by the Klaus Tschira Boost Fund via the grant Project KT43.
We express our gratitude towards these persons and institutions.
Open access
