Reinhardt cardinals and iterates of V

Abstract

Assume ZF($j$) and there is a Reinhardt cardinal, as witnessed by the elementary embedding $j:V\to V$. We investigate the linear iterates $(N_{\alpha},j_{\alpha})$ of $(V,j)$, and their relationship to $(V,j)$, forcing and definability, including that for each infinite ordinal $\alpha$, every set is set-generic over $N_{\alpha}$, but $N_{\alpha}$ is not a set-ground. Assume second order ZF. We prove that the existence of super Reinhardt cardinals and total Reinhardt cardinals is not affected by small forcing. And if $V[G]$ has a set of ordinals which is not in $V$, then $V[G]$ has no elementary embedding $j:V[G]\to M\subseteq V$ (even allowing $M$ to be illfounded).