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Kelley–Morse set theory does not prove the class Fodor principle

Volume 254 / 2021

Victoria Gitman, Joel David Hamkins, Asaf Karagila Fundamenta Mathematicae 254 (2021), 133-154 MSC: Primary 03E70, 03E25; Secondary 03E35. DOI: 10.4064/fm725-9-2020 Published online: 18 February 2021

Abstract

We show that Kelley–Morse KM set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to {\rm Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, for every $\omega \leq \lambda \leq {\rm Ord}$, it is relatively consistent with KM that there is a class function $F:{\rm Ord} \to \lambda $ that is not constant on any stationary class, and moreover $\lambda $ is the least ordinal for which such a counterexample function exists. As a corollary of this result, it is consistent with KM that there is a class $A\subseteq \omega \times {\rm Ord}$ such that each section $A_n=\{\alpha \mid (n,\alpha )\in A\}$ contains a class club, but $\bigcap _n A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma $-closed.

Authors

  • Victoria GitmanMathematics Program
    CUNY Graduate Center
    The City University of New York
    365 Fifth Avenue
    New York, NY 10016, U.S.A.
    URL: https://victoriagitman.github.io/
    e-mail
  • Joel David HamkinsProfessor of Logic
    Oxford University
    & Sir Peter Strawson Fellow
    University College
    Oxford
    High Street
    Oxford, OX1 4BH, UK
    URL: http://jdh.hamkins.org
    e-mail
  • Asaf KaragilaSchool of Mathematics University of East Anglia
    Norwich, NR4~7TJ, UK
    URL: http://karagila.org/
    e-mail

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