Partial choice functions for families of finite sets
Logic
2011-12-13 v2
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.
Cite
@article{arxiv.0808.0535,
title = {Partial choice functions for families of finite sets},
author = {Eric J. Hall and Saharon Shelah},
journal= {arXiv preprint arXiv:0808.0535},
year = {2011}
}
Comments
Early version submitted to Fundamenta Mathematicae (16 June 2008), later withdrawn, in order to rewrite with more general results. New version submitted Dec 2011