Reinhardt cardinals and iterates of V

Abstract

Assume ZF(j) and there is a Reinhardt cardinal, as witnessed by the elementary embedding j:V→V. We investigate the linear iterates (Nα,jα) of (V,j), and their relationship to (V,j), forcing and definability, including that for each infinite α, every set is set-generic over Nα, but Nα is not a set-ground.Assume Morse-Kelley set theory without the Axiom of Choice. 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]→M⊆V (even allowing M to be illfounded).