### 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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