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 $2^{\mathbb{N}}$ is Heine-Borel compactness: For any open cover of $2^{\mathbb{N}}$, there is a finite sub-cover. A natural question is: How hard is it to compute such a finite sub-cover? We make this precise by analyzing the complexity of functionals that given any $g:2^{\mathbb{N}}\rightarrow \mathbb{N}$, output a finite sequence $\langle f_0 , \dots, f_n\rangle $ in $2^{\mathbb{N}}$ such that the neighbourhoods defined from $\bar{f_i}g(f_i)$ for $i\leq n$ form a cover of Cantor space. (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 analyze the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics. The study of (T.1) gives rise to 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 and that our study of these topics is `holistic' in nature: results in computability theory give rise to results in Nonstandard Analysis and vice versa.