Reverse mathematics and equivalents of the axiom of choice

TL;DR

This paper investigates the reverse mathematical strength of countable maximality principles related to the axiom of choice, revealing a spectrum of logical equivalences and implications within second-order arithmetic.

Contribution

It identifies the precise logical strengths of several choice-related principles and their variations in the framework of reverse mathematics, including their relation to known systems like , , and .

Findings

Principles range from equivalent to  to weaker than .

Some principles are incomparable with .

A specific choice principle lies below  and implies .

Abstract

We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a $\subseteq$-maximal subfamily with the finite intersection property and the principle asserting that if $P$ is a property of finite character then every set has a $\subseteq$-maximal subset of which $P$ holds. We show that these principles and their variations have a wide range of strengths in the context of second-order arithmetic, from being equivalent to $\mathsf{Z}_2$ to being weaker than $\mathsf{ACA}_0$ and incomparable with $\mathsf{WKL}_0$. In particular, we identify a choice principle that, modulo $\Sigma^0_2$ induction, lies strictly below the atomic model theorem principle $\mathsf{AMT}$ and implies the omitting partial types principle $\mathsf{OPT}$.