On the consistency of ZF with an elementary embedding from V(λ+2) into V(λ+2)

Schlutzenberg Farmer

Other scientific publication

Abstract

This paper shows that assuming the consistency of the strongest large cardinal axioms traditionally studied under the Axiom of Choice, Kunen's proof that there is no elementary embedding j:Vλ+2→Vλ+2 cannot be carried out in ZF. Recall that I0,λ is the assertion that λ is a limit ordinal and there is an elementary embedding j:L(Vλ+1)→L(Vλ+1) with critical point < λ. This hypothesis is usually studied assuming ZFC holds in the full universe V, but we assume only ZF. We show, assuming ZF+I0,λ, that there is a proper class transitive inner model M containing Vλ+1 and modelling the theory ZF+I0,λ+"there is an elementary embedding j:Vλ+2→Vλ+2". By employing the results of the papers "Periodicity in the cumulative hierarchy" and "Even ordinals and the Kunen inconsistency", we also show that this generalizes to all even ordinals λ. In the case that λ is a limit and λ-DC holds in V, then the model M constructed also satisfies λ-DC. We also show that if ZFC+I0,λ is consistent, then it does not imply the existence of Vλ+1#. Likewise, if ZF+"λ is an even ordinal and j:L(Vλ+1)→L(Vλ+1) is elementary with critical point