On The Strength of Two Recurrence Theorems

TL;DR

This paper analyzes the logical strength of two recurrence theorems in topological dynamics using reverse mathematics, revealing a novel intermediate position of one theorem and its implications for the hierarchy of mathematical principles.

Contribution

It demonstrates that the existence of an almost periodic point is strictly between WKL and ACA in logical strength, introducing a new subclass of PA degrees.

Findings

Existence of an almost periodic point is between WKL and ACA.

The theorem is conservative over RCA_0 for Pi^1_1 sentences.

Introduces a new upwards-closed subclass of PA degrees.

Abstract

This paper uses the framework of reverse mathematics to investigate the strength of two recurrence theorems of topological dynamics. It establishes that one of these theorems, the existence of an almost periodic point, lies strictly between WKL and ACA (working over RCA_0). This is the first example of a theorem with this property. It also shows the existence of an almost periodic point is conservative over RCA_0 for Pi^1_1 sentences. These results establish the existence of a new upwards-closed subclass of the PA degrees