Reinhardt cardinals and iterates of V

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-gr...

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Bibliographic Details
Published in:Annals of pure and applied logic 2022-02, Vol.173 (2), p.103056, Article 103056
Main Author: Schlutzenberg, Farmer
Format: Article
Language:English
Subjects:
Axiom of choice
Definability
Forcing
HOD
Iterate
Reinhardt cardinal
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Summary: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).
ISSN:0168-0072
DOI:10.1016/j.apal.2021.103056