Applications of pcf for mild large cardinals to elementary embeddings

TL;DR

This paper explores pcf theory applications to large cardinals and elementary embeddings, establishing new results on cofinalities, cardinal arithmetic, and properties of elementary embeddings related to measurable and weakly compact cardinals.

Contribution

It introduces novel pcf results for large cardinals and elementary embeddings, extending the understanding of cardinal arithmetic and ultrafilter properties in set theory.

Findings

Established pcf results for weakly compact cardinals and singular cardinals.

Proved existence of specific cofinal sequences converging to singular cardinals.

Demonstrated properties of elementary embeddings related to measurable cardinals.

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 < pp^+_{Gamma(kappa)} (mu) there is an increasing sequence (lambda_i | 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} (frak a) | frak a subseteq Reg cap (mu^+, chi) and |frak 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 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 Gitik and Shelah (1993) [2] and the thrid to a question of D. Fremlin.