2008/08/05 by Eric J. Hall, Eric Hall, Saharon Shelah +2
Mathematics · #03E25 (Primary) 03E05 #15A03 (Secondary) #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03E05 #msc:03E25 #msc:15A03
paper · pdf · doi:10.48550/arxiv.0808.0535
published as Fund. Math. 220 No. 3 (2013) 207--216 · Early version submitted to Fundamenta Mathematicae (16 June 2008), later withdrawn, in order to rewrite with more general results. New version submitted Dec 2011
openalex publication_date 2008/08/05 · arxiv created 2011/12/11 · arxiv updated 2011/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.