Partial choice functions for families of finite sets

• Autorzy: Eric J. Hall , Saharon Shelah
• Języki publikacji: EN

Abstrakty

EN

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 $𝔽_{p}$. 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.

Kategorie tematyczne

• 03E25: Axiom of choice and related propositions
• 15A03: Vector spaces, linear dependence, rank

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2013
• Tom: 220
• Numer: 3
• Strony: 207-216

Daty

wydano

2013

Twórcy

autor

Eric J. Hall

• Department of Mathematics & Statistics, University of Missouri-Kansas City, Kansas City, MO 64110, U.S.A.

autor

Saharon Shelah

• Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
• Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, U.S.A.

Identyfikatory

DOI

10.4064/fm220-3-2