12 problems

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 $\theta$ 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 $\theta$ be a regular cardinal with $\theta \geq \omega_2$. The Mapping Reflection Principle, $\mathrm{MRP}$, is a forcing axiom concerning open stationary set mappings on coun…

Let $T$ be a countable theory and let $f_T(\kappa,\lambda)$ be the associated two-cardinal counting function for infinite cardinals $kappaleqlambda$. Adler's counting-spectrum conjec…

Solovay-sequence conjecture. Then $N\vDash\mathsf{ZF}$, $\Theta^N=\theta_\alpha$, $N$ satisfies that every cardinal $\eta\geq\Theta$ has regular successor $\eta^+$, and, for every…

Cardinal-regularity conjecture. Then

Shelah's chain-length conjecture. For every $\mu\geq\theta$, there are no chains in

Shelah's conjecture. If for some $\lambda\geq|\tau_1|$, $\operatorname{Df}(\lambda)>\operatorname{Ded}(\lambda)$, then for every $\lambda$,…

Let $\kappa$ be a singular cardinal, and let ${}^{<\kappa}\square\,2^\kappa$ denote the corresponding middle box product space. The paracompactness conjecture for singular cardinal…

Instability conjecture. If $\lambda>|T|$ is strong limit, $\lambda>\theta>\operatorname{cf}(\lambda)$, and $T$ is not definably stable, then

Let $IN=\{1,2,3,\ldots\}$, let $IN^{\star}=\{1^{\ast},2^{\ast},3^{\ast},\ldots\}$, and let $IN_{\Theta}=\{1_{\Theta},2_{\Theta},3_{\Theta},\ldots\}$. For a set $S$, write $2^S$ for…

Let $obreakbreak obreaklambda<mu$ be cardinals, and let $D$ be a $(\lambda^+,mu)$-regular ultrafilter. Decomposability-or-regularity conjecture. $D$ is either…