# Computability Theory, Nonstandard Analysis, and their connections

## TL;DR

This paper explores the deep connections between computability theory and Nonstandard Analysis through two main topics: the complexity of finite sub-covers in Cantor space and the strength of nonstandard compactness, revealing new insights in both fields.

## Contribution

It introduces a novel analysis linking the complexity of computing finite sub-covers with nonstandard compactness properties, bridging computability, Nonstandard Analysis, and Reverse Mathematics.

## Key findings

- Complexity of finite sub-covers analyzed in computability terms.
- Nonstandard compactness shown to have surprising strength in Reverse Mathematics.
- Intertwined results reveal deep connections between the two areas.

## 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.