PCF Theory and the Tukey Spectrum

Abstract

In this paper, we investigate the relationship between the Tukey order and PCF theory, as applied to sets of regular cardinals. We show that it is consistent that for all sets $A$ of regular cardinals that the Tukey spectrum of $A$, denoted $\operatorname{spec}(A)$, is equal to the set of possible cofinalities of $A$, denoted $\operatorname{pcf}(A)$; this is to be read in light of the $\mathsf{ZFC}$ fact that $\operatorname{pcf}(A)\subseteq\operatorname{spec}(A)$ holds for all $A$. We also prove results about when regular limit cardinals must be in the Tukey spectrum or must be out of the Tukey spectrum of some $A$, and we show the relevance of these for forcings which might separate $\operatorname{spec}(A)$ from $\operatorname{pcf}(A)$. Finally, we show that the strong part of the Tukey spectrum can be used in place of PCF-theoretic scales to lift the existence of Jonsson algebras from below a singular to hold at its successor. We close with a list of questions.