A weak reflection of Reinhardt by super Reinhardt cardinals

Abstract

We prove a weakened version of the reflection of Reinhardt cardinals by super Reinhardt cardinals: Let $M=(V^M,P)$ be a countable model of second order set theory $\mathsf{ZF}_2$ (with universe $V^M$ and classes $P$) which models "$\kappa$ is super Reinhardt". We show that there are unboundedly many $\mu<\kappa$ such that there is $j$ such that $(V^M,j)$ models $\mathsf{ZF}(j)+$"$\mu$ is Reinhardt, as witnessed by $j$". In particular, $j\upharpoonright X\in V^M$ for all $X\in V^M$ (but we allow $j\notin P$).