Connecting real and hyperarithmetical analysis

TL;DR

This paper explores the connection between real analysis theorems and hyperarithmetical analysis, providing numerous examples within a specific logical range and identifying new systems in the process.

Contribution

It offers the first extensive collection of real analysis theorems fitting within hyperarithmetical analysis and introduces new systems in this logical framework.

Findings

Examples of real analysis theorems within hyperarithmetical range

Identification of new hyperarithmetical systems

Application of the framework to Jordan decomposition and metric spaces

Abstract

Going back to Kreisel in the Sixties, hyperarithmetical analysis is a cluster of logical systems just beyond arithmetical comprehension. Only recently natural examples of theorems from the mathematical mainstream were identified that fit this category. In this paper, we provide many examples of theorems of real analysis that sit within the range of hyperarithmetical analysis, namely between the higher-order version of $\Sigma_1^1$-AC$_0$ and weak-$\Sigma_1^1$-AC$_0$, working in Kohlenbach's higher-order framework. Our example theorems are based on the Jordan decomposition theorem, unordered sums, metric spaces, and semi-continuous functions. Along the way, we identify a couple of new systems of hyperarithmetical analysis.