Applications of pcf for mild large cardinals to elementary embeddings

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Abstract

The following pcf results are proved:. 1. Assume that κ>א0 is a weakly compact cardinal. Let μ>2κ be a singular cardinal of cofinality κ. Then for every regular λ<ppΓ(κ)+(μ) there is an increasing sequence 〈λi|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)|=|jW(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. © 2013 Elsevier B.V.

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APA

Gitik, M., & Shelah, S. (2013). Applications of pcf for mild large cardinals to elementary embeddings. Annals of Pure and Applied Logic, 164(9), 855–865. https://doi.org/10.1016/j.apal.2013.03.002

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