Computability Theory, Nonstandard Analysis, and Their Connections

Abstract

We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related.(T.1) A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output a finite sequence $\langle f_0 , \ldots ,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left( {f_i } \right)$ for $i \le n$ form a covering of $2^ $.(T.2) A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.Our study of (T.1) yields exotic objects in computability theory, while (T.2) leads to surprising results in Reverse Mathematics. We stress that (T.1) and (T.2) are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa.

Naïve Infinitesimal Analysis: Its Construction and Its Properties

“…In a wide sense, the work on this article can also be seen as a contribution to bridge (the antipodes) constructive analysis and nonstandard analysis. This problem has been extensively (and intensively) discussed in the past few years (see for example [40,41,42,8,43,31]).…”

Computability Theory, Nonstandard Analysis, and their connections

“…, f n such that the neighbourhoods defined from f i g(f i ) for i ≤ n form a cover of Cantor space; almost as a by-product, Θ(g)(1) can then be chosen to be the maximal value of g(f i ) + 1 for i ≤ n. We stress that g 2 in SCF(Θ) may be discontinuous and that Kohlenbach has argued for the study of discontinuous functionals in higher-order RM (See Section 2.2). As it turns out, the functional Θ is intimately connected to Tao's notion of metastability, as explored in [39,49].…”

“…In particular, a fully detailed proof requires a tailor-made modification of the Kleene schemes S1-S9, and this would take up more space than available in this Appendix. A detailed proof will appear in [39].…”

Computability and Non-monotone induction

“…It is worth noticing that the set-theoretical language mostly used at the time is of third order, while coding is needed to capture the same concepts in second order arithmetic (SOA). In a series of papers [14][15][16][17][18][19][20], Sam Sanders and the author have investigated the logical and computability strength of some of the results using such tools, when expressed in a language close to how it was originally done.…”

“…In the terminology of today, Borel constructed a functional taking the map x → O x as the argument and yielding a finite subcovering as the value. The definition of this functional is by transfinite recursion, building up finite subcoverings of larger and larger closed subintervals, a construction that can be viewed as a simultaneous non-monotone inductive definition of Dedekind cuts for numbers c ≤ b and finite subcoverings of each closed interval [a, d] for d < c. In [14], a realiser Θ 0 of the uncountable Heine-Borel theorem (HBU) is defined. This realiser selects a finite set x 1 , .…”