Open sets in computability theory and Reverse Mathematics

TL;DR

This paper investigates how additional data and structure on open sets affect their computational and logical properties, revealing nuanced differences in classical theorems within higher-order computability and reverse mathematics.

Contribution

It introduces a detailed analysis of open sets with extra structure, examines their impact on foundational theorems, and proposes the new $\,\Delta$-functional with unique computational features.

Findings

Open sets with extra data alter the computability of classical theorems.

Theorems like Baire, Heine, and Tietze behave differently under structured open sets.

The $\,\Delta$-functional provides a refined tool for representing open sets in computational frameworks.

Abstract

To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this paper, what the influence of this extra data and structure is on the logical and computational properties of basic theorems pertaining to open sets. To answer this question, we study various basic theorems of analysis, like the Baire category, Heine, Heine-Borel, Urysohn, and Tietze theorems, all for open sets given by their (third-order) characteristic functions. Regarding computability theory, the objects claimed to exist by the aforementioned theorems undergo a shift from `computable' to `not computable in any type two functional', following Kleene's S1-S9. Regarding Reverse Mathematics, the latter's so-called Main Question, namely which set existence axioms are necessary for proving a given theorem, does not have a unique or unambiguous answer for the aforementioned theorems, working in Kohlenbach's higher-order framework. A finer study of representations of open sets leads to the new `$\Delta$-functional' which has unique (computational) properties.