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Partial choice functions for families of finite sets

Volume 220 / 2013

Eric J. Hall, Saharon Shelah Fundamenta Mathematicae 220 (2013), 207-216 MSC: Primary 03E25; Secondary 03E25, 15A03. DOI: 10.4064/fm220-3-2

Abstract

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.

Authors

  • Eric J. HallDepartment of Mathematics & Statistics
    University of Missouri–Kansas City
    Kansas City, MO 64110, U.S.A.
    e-mail
  • Saharon ShelahEinstein 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.
    e-mail

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