# Questions tagged [pcf-theory]

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19 questions
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### “Towers” on singular cardinals with countable cofinality

Let $\lambda$ be a singular cardinal of countable cofinality. Is there necessarily a sequence $\{A_\alpha\mid\alpha<\lambda^+\}$ of countable subsets of $\lambda$, such that $\alpha<\beta$ if ...
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### Possible cofinalities of cuts of ultraproducts

Suppose $\kappa$ is a regular cardinal, $\bar{\mu}=(\mu_i: i<\kappa)$ is an increasing sequence of regular cardinals ($>\kappa$) and set $pcut(\bar \mu)=\{ (\lambda_1, \lambda_2):$ for some ...
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### Consistency strength of the failure of Shelah's Strong Hypothesis (SSH)

Some known facts about SSH (Shelah's Strong Hypothesis): i) "$0^\sharp$ does not exist" implies SSH. ii) SSH implies SCH (Singular Cardinal Hypothesis). iii) The failure of SCH is equiconsistent ...
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### Bounds on $\max \mathrm{pcf}(A)$ if $\Pi A$ is big

For concreteness, let $A = \{\aleph_n : n < \omega\}$. We know $\max \mathrm{pcf}(A) \in [\aleph_{\omega+1},\Pi A]$. My question is, if $\Pi A$ is big (say, $\aleph_{\omega_1+1}$), then which ...
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### Some Pcf Theory

Let $pcf(a)$ denote the set of regular cardinals such that $J_{\leq \lambda} - J_{<\lambda} \neq \emptyset$ and let $maxpcf(a)$ denote the maximum of $pcf(a)$. The $J_{\leq \lambda}$ are the usual ...