Reverse mathematics and properties of finite character

TL;DR

This paper investigates the reverse mathematical strength of the principle that every property of finite character has a maximal subset, revealing its equivalence to the axiom of choice in set theory and analyzing its behavior in second-order arithmetic.

Contribution

It provides a full characterization of the principle's strength based on the formula's quantifier structure and explores its interaction with closure operators.

Findings

The principle is equivalent to the axiom of choice in set theory.

Its strength varies with the quantifier complexity of the defining formula.

It interacts with finitary and nondeterministic closure operators in specific ways.

Abstract

We study the reverse mathematics of the principle stating that, for every property of finite character, every set has a maximal subset satisfying the property. In the context of set theory, this variant of Tukey's lemma is equivalent to the axiom of choice. We study its behavior in the context of second-order arithmetic, where it applies to sets of natural numbers only, and give a full characterization of its strength in terms of the quantifier structure of the formula defining the property. We then study the interaction between properties of finite character and finitary closure operators, and the interaction between these properties and a class of nondeterministic closure operators.