Taking Reinhardt's Power Away

TL;DR

This paper investigates the non-existence of certain elementary embeddings in models of set theory lacking the Power Set axiom, and explores failures of collection in symmetric submodels of class forcings.

Contribution

It proves the non-existence of cofinal elementary embeddings under specific conditions in models without Power Set, and analyzes collection failures in symmetric submodels.

Findings

No cofinal elementary embeddings exist under the given assumptions.

Failures of collection occur in symmetric submodels of class forcings.

Results extend understanding of set-theoretic embeddings without Power Set.

Abstract

We study the notion of non-trivial elementary embeddings $j : V \rightarrow V$ under the assumption that $V$ satisfies $ZFC$ without Power Set but with the Collection Scheme. We show that no such embedding can exist under the additional assumption that it is cofinal and either $V_{\textrm{crit}(j)}$ is a set or that the Dependent Choice Schemes holds. We then study failures of instances of collection in symmetric submodels of class forcings.