Weak diamond and pcf theory

TL;DR

This paper explores the relationship between weak diamond principles and pcf theory to establish bounds on the size of certain power sets, impacting the understanding of singular cardinals in set theory.

Contribution

It introduces new bounds on pcf sets derived from weak diamond assumptions, linking combinatorial principles to cardinal arithmetic.

Findings

Bounds on |pcf(𝔞)| from weak diamond instances

Existence of many singular cardinals with specific power set bounds

Conditions under which the bounds apply to all limit cardinals

Abstract

We obtain bounds on the cardinality of $pcf(\mathfrak{a})$ from instances of weak diamond. Consequently, under mild assumptions there are many singular cardinals of the from $\aleph_\delta$ for which $2^{\aleph_\delta}<\aleph_{(|\delta|^{+3})}$. For example, if every limit cardinal is a strong limit cardinal then this bound holds at a class of singular cardinals.