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>0 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)|aReg(μ+,χ)and|a|<μ}.

As an application we show that:

if κ is a measurable cardinal and j:VM 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.

MSC

03E35
03E45
03E55
03E04

Keywords

Measurable cardinal
Weakly compact cardinal
Elementary embedding
Cardinal arithmetic
pcf-generators
Revised GCH

Cited by (0)

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 Shelahʼs publication list.