# Sh:1013

• Gitik, M., & Shelah, S. (2013). Applications of pcf for mild large cardinals to elementary embeddings. Ann. Pure Appl. Logic, 164(9), 855–865.
• Abstract:
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<{\rm pp}^+_{\Gamma(\kappa)}(\mu) there is an increasing sequence \langle \lambda_i \mid i<\kappa \rangle of regular cardinals converging to \mu such that \lambda= {\rm 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> {\rm sup}\{ {\rm sup} {\rm pcf}_{\sigma_{*}-{ \rm complete}}(\mathfrak{a}) \mid \mathfrak{a}\subseteq {\rm Reg} \cap (\mu^+,\chi) \text{ and } | \mathfrak{a}|<\mu \}. As an application we show that:

if \kappa is a measurable cardinal and j:V \to M is the elementary embedding by a \kappa–complete ultrafilter over a measurable cardinal \kappa, then for every \tau the following holds:

1. if j(\tau) is a cardinal then j(\tau)=\tau;

2. |j(\tau)|=|j(j(\tau))|;

3. for any \kappa–complete ultrafilter W on \kappa, |j(\tau)|=|j_W(\tau)|.

The first two items provide affirmative answers to questions from [G-Sh] and the third to a question of D. Fremlin.

• published version (11p)
Bib entry
@article{Sh:1013,
author = {Gitik, Moti and Shelah, Saharon},
title = {{Applications of pcf for mild large cardinals to elementary embeddings}},
journal = {Ann. Pure Appl. Logic},
fjournal = {Annals of Pure and Applied Logic},
volume = {164},
number = {9},
year = {2013},
pages = {855--865},
issn = {0168-0072},
mrnumber = {3056300},
mrclass = {03E55 (03E04 03E35 03E45)},
doi = {10.1016/j.apal.2013.03.002},
note = {\href{https://arxiv.org/abs/1307.5977}{arXiv: 1307.5977}},
arxiv_number = {1307.5977}
}