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PCF theory and the Tukey spectrum

Volume 265 / 2024

Thomas Gilton Fundamenta Mathematicae 265 (2024), 15-33 MSC: Primary 03E04; Secondary 03E05 DOI: 10.4064/fm254-1-2024 Published online: 26 April 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.

Authors

  • Thomas GiltonDepartment of Mathematics
    University of Pittsburgh
    The Dietrich School of Arts and Sciences
    Pittsburgh, PA 15260, USA
    e-mail

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