ZFC without power set II: Reflection strikes back

TL;DR

This paper explores the failure of dependent choice principles in models of ZFC without power set, introduces the concept of big proper classes, and constructs models demonstrating various independence results related to these choice schemes.

Contribution

It develops a framework for separating dependent choice schemes of different lengths and constructs models where these principles fail or hold in specific ways.

Findings

Models of ZFC- with non-big proper classes are constructed.

Dependent choice of length ω can fail even with many unbounded cardinals.

Existence of certain elementary embeddings does not imply the full von-Neumann hierarchy.

Abstract

The theory ZFC implies the scheme 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 ZFC$^-$ (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 ZFC$^-$ by considering the notion of big proper classes. A proper class is said to be big if it surjects onto every non-zero ordinal. We shall see that if one assumes the scheme of dependent choices of any arbitrary set length then every proper class is indeed big. However, by building on work of Zarach, we provide a general framework for separating dependent choice schemes of various lengths by producing models of ZFC$^-$ with proper classes that are not big. Using a similar idea, we then extend the earlier result by producing a model of 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 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, that in such a situation $\mathcal{P}(\omega)$ can be a proper class.