Partial choice functions for families of finite sets

Eric J. Hall ¹   ; Saharon Shelah ²

¹ Department of Mathematics & Statistics University of Missouri–Kansas City Kansas City, MO 64110, U.S.A.
² Einstein Institute of Mathematics Edmond J. Safra Campus, Givat Ram The Hebrew University of Jerusalem Jerusalem, 91904, Israel and Department of Mathematics Rutgers University New Brunswick, NJ 08854, U.S.A.

Fundamenta Mathematicae, Tome 220 (2013) no. 3, pp. 207-216

Résumé

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.

Détail

DOI : 10.4064/fm220-3-2

Keywords: integer every countable set m element sets has infinite partial choice function strong enough prove every countable set m element sets has choice function answering question actually slightly stronger result obtained independence result where prime obtained permutation fraenkel mostowski model zfa which set atoms urelements has structure vector space finite field mathbb atoms eliminated citing embedding theorem pincus where prime suitable permutation models built models prime cases