10 problems

Shelah's Strong Hypothesis. For every singular cardinal $\sigma$,

Shelah's PCF conjecture.

Assume that $\lambda_n$ is the $n$-th inaccessible cardinal, let $\kappa=\sum_{n<\omega}\lambda_n$, and consider a forcing notion $\mathbb P$. Shelah's forcing conjecture for pseud…

Let $\aleph_\delta$ be the first fix point, meaning the first ordinal index satisfying $\aleph_\delta=\delta$, and hence having cofinality $\aleph_0$. Shelah's first-fix-point pseu…

Let $\mu$ be a limit of limit cardinals, and let $\lambda<\mu$ range over arbitrarily large regular cardinals. Write $\forall^*$ for “for every large enough.” Shelah's pcf conjectu…

Let $\theta$ be any uncountable cardinal. Shelah's consistency conjecture for pseudopowers. It is consistent that, for some cardinal $\lambda$, both of the following hold: (A)…

For a singular cardinal $\theta$, let $\rm pp(\theta)$ denote its pseudopower, as defined by the supremum of true cofinalities of reduced products of regular cardinals below…

A cardinal is called a revision cardinal when it has the revision property developed above. Revision Hypothesis. Any limit cardinal is a revision cardinal. This hypothesis is prese…

Shelah's conjecture. A progressive set of regular cardinals can never have an inaccessible accumulation point.

Let $\theta$ and $\mu$ be cardinals with $\mu>\theta$. Consider the partial order of functions ${}^{(\theta^{+3})}\mu$ ordered modulo the ideal $[\theta^{+3}]^{\leq\theta}$ of subs…