Partition models, Permutations of infinite sets without fixed points, and weak forms of AC

TL;DR

This paper explores the relationships between weak choice principles and various mathematical statements in ZF and ZFA, establishing equivalences and analyzing their status in permutation models.

Contribution

It introduces new equivalences between weak choice principles and classical theorems, and studies their validity in permutation models and finite partition models.

Findings

Equivalence of certain graph and field properties with weak choice principles.

Van Douwen's Choice Principle holds in specific permutation models.

The principle about infinite Hausdorff spaces relates to weak choice assumptions.

Abstract

We study new relations of the following statements with weak choice principles in ZF and ZFA. 1. There does not exist an infinite Hausdorff space X such that every infinite subset of X contains an infinite compact subset. 2. If a field has an algebraic closure then it is unique up to isomorphism. 3. For every infinite set X, there exists a permutation of X without fixed points. Moreover, we prove that the principle ``Any infinite locally finite connected graph has a spanning m-bush for any even integer m greater than or equal to 4'' is equivalent to K\H{o}nig's Lemma in ZF. We also study the new status of different weak choice principles in the finite partition model (a type of permutation model) introduced by B.B. Bruce in 2016. Further, we prove that Van Douwen's Choice Principle holds in two recently constructed known permutation models.