Partial choice functions for families of finite sets
Volume 220 / 2013
Fundamenta Mathematicae 220 (2013), 207-216
MSC: Primary 03E25; Secondary 03E25, 15A03.
DOI: 10.4064/fm220-3-2
Abstract
Let $m\ge 2$ be an integer. We show that ZF $+$ “Every countable set of $m$-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of $m$-element sets has a choice function, answering an open question from . (Actually a slightly stronger result is obtained.) The independence result in the case where $m=p$ is prime is obtained by way of a permutation (Fraenkel–Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure of a vector space over the finite field $\mathbb {F}_{p}$. The use of atoms is then eliminated by citing an embedding theorem of Pincus. In the case where $m$ is not prime, suitable permutation models are built from the models used in the prime cases.
