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ZFC without power set II: reflection strikes back

Volume 264 / 2024

Victoria Gitman, Richard Matthews Fundamenta Mathematicae 264 (2024), 149-178 MSC: Primary 03E30; Secondary 03E35, 03E40 DOI: 10.4064/fm206-11-2023 Published online: 14 February 2024


$\mathrm {ZFC}$ implies that for every cardinal $\delta $ we can make $\delta $-many dependent choices over any definable relation without terminal nodes. Friedman, the first author, and Kanovei constructed a model of $\mathrm {ZFC}^{-}$ ($\mathrm {ZFC}$ without power set) with largest cardinal $\omega $ in which this principle fails for $\omega $-many choices. In this article we study failures of dependent choice principles over $\mathrm {ZFC}^{-}$.

Building upon work of Zarach, we provide a general framework for separating dependent choice schemes of various lengths by producing models of $\mathrm {ZFC}^{-}$. Using a similar idea, we then extend the earlier result by producing a model of $\mathrm {ZFC}^{-}$ in which there are unboundedly many cardinals but the scheme of dependent choices of length $\omega $ still fails.

Finally, the second author has proven that a model of $\mathrm {ZFC}^{-}$ cannot have a non-trivial, cofinal, elementary self-embedding for which the von Neumann hierarchy exists up to its critical point. We answer a related question posed by the second author by showing that the existence of such an embedding need not imply the existence of any non-trivial fragment of the von Neumann hierarchy. In particular, in such a situation $\mathcal {P}(\omega )$ can be a proper class.


  • Victoria GitmanMathematics Program
    CUNY Graduate Center
    The City University of New York
    New York, NY 10016, USA
  • Richard MatthewsUniv. Paris Est Créteil, LACL
    F-94010 Créteil, France

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