Let m ≥ 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 . 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-3-2,
author = {Eric J. Hall and Saharon Shelah},
title = {Partial choice functions for families of finite sets},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {207-216},
zbl = {1271.03069},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-3-2}
}
Eric J. Hall; Saharon Shelah. Partial choice functions for families of finite sets. Fundamenta Mathematicae, Tome 220 (2013) pp. 207-216. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm220-3-2/