6 problems

Let $\lambda$ and $\kappa$ be cardinals, with $\lambda=\lambda^{<\lambda}>\kappa^+$, $\kappa=\operatorname{cf}(\kappa)$, and $\lambda\ne\aleph_1$. The weaker alternative allows…

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)…

Let ${\mathfrak a}$ be a set of regular cardinals such that every member of ${\mathfrak a}$ is greater than $|{\mathfrak a}|$. Shelah's pcf boundedness conjecture. For no inaccessi…

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…