PCF theory and the Tukey spectrum
Volume 265 / 2024
Abstract
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, the Tukey spectrum of $A$, denoted $\mathrm{spec}(A)$, is equal to the set of possible cofinalities of $A$, denoted $\mathrm{pcf}(A)$; this is to be read in light of the $\mathsf{ZFC}$ fact that $\mathrm{pcf}(A)\subseteq \mathrm{spec}(A)$ 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 $\mathrm{spec}(A)$ from $\mathrm{pcf}(A)$. 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 Jónsson algebras from below a singular to hold at its successor. We close with a list of questions.
