Applications of pcf for mild large cardinals to elementary embeddings

The following pcf results are proved:. 1. Assume that κ>א₀ is a weakly compact cardinal. Let μ>2^κ be a singular cardinal of cofinality κ. Then for every regular λ<ppΓ(κ)+(μ) there is an increasing sequence 〈λ_i|i<κ〉 of regular cardinals converging to μ such that λ=tcf(∏i<κλi,<Jκbd).2. Let μ be a strong limit cardinal and θ a cardinal above μ. Suppose that at least one of them has an uncountable cofinality. Then there is σ_*<μ such that for every χ<θ the following holds:θ>sup{suppcfσ*-complete(a)|a⊆Reg∩(μ+,χ)and|a|<μ}. As an application we show that:. if κ is a measurable cardinal and j: V→ M is the elementary embedding by a κ-complete ultrafilter over κ, then for every τ the following holds:. 1.if j(τ) is a cardinal then j(τ)=τ;2.|j(τ)|=|j(j(τ))|;3.for any κ-complete ultrafilter W on κ, |j(τ)|=|j_W(τ)|. The first two items provide affirmative answers to questions from Gitik and Shelah (1993) [2] and the third to a question of D. Fremlin.