A weak reflection of Reinhardt by super Reinhardt cardinals

Farmer Schlutzenberg

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Abstract

This note deals with large cardinal principles (strong set theoretic axioms of infinity) which are beyond the level consistent with the Axiom of Choice. We give a partial answer to a question of Bagaria, Koellner and Woodin, proving the following weakened version of the reflection of Reinhardt cardinals by super Reinhardt cardinals: Let M = (V, P) be a countable model of second order set theory ZF2 (with universe V and classes P) which models "κ is super Reinhardt". We show that there are unboundedly many μ < κ such that there is j such that (V, j) models ZF(j) + "μ is Reinhardt, as witnessed by j". In particular, j↾X ∈ V for all X ∈ V (but we allow j ∉ P).