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Embedding orders into the cardinals with $\mathsf {DC}_{\kappa} $

Volume 226 / 2014

Asaf Karagila Fundamenta Mathematicae 226 (2014), 143-156 MSC: Primary 03E25; Secondary 03E35. DOI: 10.4064/fm226-2-4

Abstract

Jech proved that every partially ordered set can be embedded into the cardinals of some model of $\mathsf {ZF}$. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of $\mathsf {ZF}+\mathsf {DC}_{<\kappa }$ for any regular $\kappa $. We use this theorem to show that for all $\kappa $, the assumption of $\mathsf {DC}_\kappa $ does not entail that there are no decreasing chains of cardinals. We also show how to extend the result to and embed into the cardinals a proper class which is definable over the ground model. We use this extension to give a large-cardinals-free proof of independence of the weak choice principle known as $\mathsf {WISC}$ from $\mathsf {DC}_\kappa $.

Authors

  • Asaf KaragilaEinstein Institute of Mathematics
    Edmond J. Safra Campus
    The Hebrew University of Jerusalem
    Givat Ram, Jerusalem, 91904, Israel
    e-mail

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