Applications of pcf for mild large cardinals to elementary embeddings

by Gitik and Shelah. [GiSh:1013]
Annals Pure and Applied Logic, 2013
The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu >2^kappa be a singular cardinal of cofinality kappa . Then for every regular lambda < pp^+_{Gamma(kappa)}(mu) there is an increasing sequence < lambda_i mid i< kappa > of regular cardinals converging to mu such that lambda = tcf (prod_{i< kappa} lambda_i, <_{J^{bd}_{kappa}}) . 2. Let mu be a strong limit cardinal and theta a cardinal above mu . Suppose that at least one of them has an uncountable cofinality. Then there is sigma_*< mu such that for every chi < theta the following holds: theta > sup {sup pcf_{sigma_{*}- complete}({a}) mid {a} subseteq Reg cap (mu^+, chi) and | {a}|< mu}. As an application we show that: if kappa is a measurable cardinal and j:V-> M is the elementary embedding by a kappa --complete ultrafilter over a measurable cardinal kappa, then for every tau the following holds: begin {enumerate} item if j(tau) is a cardinal then j(tau)= tau ; item |j(tau)|=|j(j(tau))| ; item for any kappa --complete ultrafilter W on kappa, quad |j(tau)|=|j_W(tau)| . end {enumerate} The first two items provide affirmative answers to questions from [G-Sh] and the third to a question of D. Fremlin.


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