PCF Theory and the Tukey Spectrum
| Autor: | Gilton, Thomas |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: Comments welcome, via email |
| Databáze: | arXiv |
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