# 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. arXiv: 1307.5977 DOI: 10.1016/j.apal.2013.03.002 MR: 3056300 -
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:

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

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

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}, doi = {10.1016/j.apal.2013.03.002}, mrclass = {03E55 (03E04 03E35 03E45)}, mrnumber = {3056300}, mrreviewer = {Hiroshi Sakai}, doi = {10.1016/j.apal.2013.03.002}, note = {\href{https://arxiv.org/abs/1307.5977}{arXiv: 1307.5977}}, arxiv_number = {1307.5977} }