Nonabsoluteness of elementary embeddings

Extract

If κ is a measurable cardinal, let us say that a measure on κ is a κ-complete nonprincipal ultrafilter on κ. If U is a measure on κ, let j_U be the canonical elementary embedding of V into its Ultrapower Ult_U(V). If x is a set, say that U moves x when j_U(x) ≠ x; say that κ moves x when some measure on κ moves x. 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 ‹U_n: n ∈ ω› be a sequence of measures on a strictly increasing sequence ‹κ_n: n ∈ ω› of measurable cardinals. Let U = ‹ W_α: α < ω²›, where W_{ωm + n} = U_m (m, n ∈ ω). Then, for each θ in Ult_U(V), if E is the (minimal) support of θ in Ult_U(V), then, for all m ∈ ω, U_m moves θ iff E ∩ [ωm, ω(m + 1)) ≠ ∅.