TY - JOUR

T1 - Applications of pcf for mild large cardinals to elementary embeddings

AU - Gitik, Moti

AU - Shelah, Saharon

N1 - Funding Information:
E-mail addresses: [email protected] (M. Gitik), [email protected] (S. Shelah). URL: http://www.math.tau.ac.il/~gitik (M. Gitik). 1 We are grateful to Menachem Magidor and the referee of the paper for their comments. Gitik was partially supported by ISF grant 234/08. 2 Shelah was partially supported by ISF grant 1053/11. This is paper 1013 on Shelah’s publication list.

PY - 2013/9

Y1 - 2013/9

N2 - 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 λi|i<κ〉 of regular cardinals converging to μ such that λ=tcf(∏i<κλi,*<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.

AB - 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 λi|i<κ〉 of regular cardinals converging to μ such that λ=tcf(∏i<κλi,*<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.

KW - Cardinal arithmetic

KW - Elementary embedding

KW - Measurable cardinal

KW - Pcf-generators

KW - Revised GCH

KW - Weakly compact cardinal

UR - http://www.scopus.com/inward/record.url?scp=84878120964&partnerID=8YFLogxK

U2 - 10.1016/j.apal.2013.03.002

DO - 10.1016/j.apal.2013.03.002

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AN - SCOPUS:84878120964

SN - 0168-0072

VL - 164

SP - 855

EP - 865

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 9

ER -