Applications of pcf for mild large cardinals to elementary embeddings

作者：Gitik Moti^*; Shelah Saharon

来源：Annals of Pure and Applied Logic, 2013, 164(9): 855-865.

DOI：10.1016/j.apal.2013.03.002

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

The following pcf results are proved:
1. Assume that K > N-0 is a weakly compact cardinal. Let mu > 2(kappa) be a singular cardinal of cofinality kappa. Then for every regular lambda < pp(r(kappa))(+)(mu) there is an increasing sequence <lambda(i) vertical bar i < K > of regular cardinals converging to mu such that lambda = tcf(Pi(i<kappa) lambda(1) < (J kappa bd)).
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) vertical bar a subset of Reg boolean AND (mu(+), chi) and vertical bar a vertical bar < 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 kappa, then for every tau the following holds:
1. if j(tau)is a cardinal then j(tau) = tau;
2. vertical bar j(tau)vertical bar= vertical bar j(j(tau))vertical bar;
3. for any kappa-complete ultrafilter W on kappa, vertical bar j(tau)vertical bar = vertical bar j(W)(tau)vertical bar.
The first two items provide affirmative answers to questions from Gitik and Shelah (1993) [2] and the third to a question of D. Fremlin.

• 出版日期2013-9