Nonabsoluteness of elementary embeddings

Abstract

Ifκis a measurable cardinal, let us say that a measure onκis aκ-complete nonprincipal ultrafilter onκ. IfUis a measure onκ, letjUbe the canonical elementary embedding ofVinto its Ultrapower UltU(V). Ifxis a set, say thatUmovesxwhenjU(x)≠x; say thatκmovesxwhen some measure onκmovesx. Recall Kunen's lemma (see [K]): “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof (see [K]) and Fleissner's proof (see [KM, III, §10]) are essentially nonconstructive.The following proposition can be proved by using elementary facts about iterated ultrapowers.Proposition.Let ‹Un: n ∈ ω› be a sequence of measures on a strictly increasing sequence ‹κn: n ∈ ω› of measurable cardinals. Let U = ‹ Wα: α < ω2›, where Wωm + n= Um(m, n ∈ ω). Then, for each θ inUltU(V),if E is the (minimal) support of θ inUltU(V),then, for all m ∈ ω, Ummoves θ iff E ∩ [ωm, ω(m + 1))≠ ∅.