Partial choice functions for families of finite sets

TL;DR

The paper demonstrates that certain weak forms of the Axiom of Choice related to partial choice functions for finite sets are insufficient to prove the existence of choice functions for all countable sets of fixed finite size, using permutation models.

Contribution

It introduces new independence results showing the limitations of ZF plus partial choice principles for finite sets, especially for non-prime sizes.

Findings

Partial choice principles are insufficient for certain finite set families

Permutation models demonstrate independence results

Results differ between prime and non-prime set sizes

Abstract

Let m>2 be an integer. We show that ZF + "For every integer n, Every countable family of non-empty sets of cardinality at most n has an infinite partial choice function" is not strong enough to prove that every countable set of m-element sets has a choice function. In the case where m=p is prime, to obtain the independence result we make use of a permutation model in which the set of atoms has the structure of a vector space over the field of p elements. When m is non-prime, a suitable permutation model is built from the models used in the prime cases.