ZFC without power set II: reflection strikes back
Volume 264 / 2024
Abstract
$\mathrm {ZFC}$ 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.
