On the consistency of ZF with an elementary embedding from Vλ+2 into Vλ+2

Abstract

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal λ and nontrivial elementary embedding j : Vλ+2 → Vλ+2. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered. I0,λ is the assertion, introduced by Hugh Woodin, that λ is an ordinal and there is an elementary embedding j : L(Vλ+1) → L(Vλ+1) with critical point < λ. And I0 asserts that I0,λ holds for some λ. The axiom I0 is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe V (in which case λ must be a limit ordinal), but we assume only ZF. We prove, assuming ZF + I0,λ + “λ is an even ordinal”, that there is a proper class transitive inner model M containing Vλ+1 and satisfying ZF + I0,λ + “there is an elementary embedding k : Vλ+2 → Vλ+2”; in fact we will have k ⊆ j, where j witnesses I0,λ in M. This result was first proved by the author under the added assumption that Vλ#+1 exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also λ is a limit ordinal and λ-DC holds in V , then the model M will also satisfy λ-DC. We show that ZFC + “λ is even” + I0,λ implies A# exists for every A ∈ Vλ+1, but if consistent, this theory does not imply Vλ#+1 exists.