Taking Reinhardt's Power Away

Abstract

We study the notion of non-trivial elementary embeddings j : V → 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 crit(j ) is a set or that the Reflection Principle holds. We then study failures of instances of collection in symmetric submodels of class forcings.

How strong is a Reinhardt set over extensions of CZF?

“…Note that Ziegler [24] uses the term set cofinality to denote our notion of cofinality. However, we will use the term cofinality to harmonize the terminology with that of Matthews [18]. 4.…”

“…Proof. Matthews [18] showed that ZFC − with the Reflection principle proves there is no elementary embedding j : V ≺ V which is cofinal. By Theorem 2.15, RRS implies the Reflection principle, so we have a contradiction.…”

“…For example, Schultzenberg [21] showed that ZF with an elementary embedding j : V λ+2 ≺ V λ+2 is consistent if ZFC with I 0 is. Matthews [18] proved that Reinhardt cardinal is compatible with ZFC − if we assume the consistency of ZFC with I 1 .…”