Taking Reinhardt's Power Away

Abstract

We study the notion of non-trivial elementary embeddings $j:V\to V$ 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 $V_{\text{crit}(j)}$ is a set or that the Dependent Choice Schemes holds. We then study failures of instances of collection in symmetric submodels of class forcings.

20 pages. Expanded the proof of Theorem 2.2, corrected the notation in Section 6