Taking Reinhardt's Power Away

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.