Abstract
Modulo the existence of large cardinals, there is a model of set theory in which, for some set B of regular cardinals, the sequence \(\langle \textrm{pcf}^\alpha (B): \alpha \in \textrm{Ord}\rangle \) is strictly increasing. The result answers a question from Tsukuura (Tsukuba J Math 45(2):83–95, 2021).
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Notes
This can always be done by forcing below a condition, which forces each element of the Radin club to be in \(A''\).
Indeed one can say more. Given a condition \(p \in \mathbb {R}\) such that \(\alpha \) and \(\beta \) appear as successive points in p, one can factor \(\mathbb {R}/p\) as \(\big (\mathbb {R}^{< \alpha }/ p(<\alpha )\big ) \times \big (\prod _{i<4} \textrm{Add}(\beth _i(\alpha ), \beth _{i+1}(\alpha )) \times \textrm{Add}(\beth _4(\alpha ), \beta ) \big ) \times \big (\mathbb {R}^{>\beta } / p(>\beta ) \big )\). Note that this representation does not depend on the club C, but once we have C, given \(\alpha \in C\), \(\beta =\min (C {\setminus } (\alpha +1))\) and \(p \in {\textbf {G}}\), we can always find an extension q of p such that \(q \in {\textbf {G}}_{\mathbb {R}}\) and \(\alpha \) and \(\beta \) appear in q.
See (2-2)(c).
Indeed, it is forced to have more closure properties, but \(\zeta _{\beta , k}^+\)-directed closedness is sufficient for us.
Note that the Cohen forcing notions on both iterands of \({\mathbb {K}}\) are defined in the same universe, so the equivalence follows from the fact that \({\textrm{Add}}(\zeta _{\beta , j}, \zeta _{\beta , j+1}) \times {\textrm{Add}}(\zeta _{\beta , j}, \zeta _{\beta , j+1}) \simeq {\textrm{Add}}(\zeta _{\beta , j}, \zeta _{\beta , j+1}).\)
When \(k=0\), the forcing becomes the trivial forcing, so the conclusion is still clear, and we can for example take \(\zeta ^+_{\beta , -1}=\aleph _1.\)
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The author’s research has been supported by a grant from IPM (No. 1401030417). He thanks the referee of the paper for his many useful remarks and suggestions that improved the presentation of the paper.
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Golshani, M. Unlimited accumulation by Shelah’s PCF operator. Period Math Hung 88, 273–280 (2024). https://doi.org/10.1007/s10998-023-00553-2
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DOI: https://doi.org/10.1007/s10998-023-00553-2