15 problems

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 $R$ be a binary relation. Write $\fulldiamond$ for the proper forcing axiom appearing in the source, and let $\beta\to(\beta_1;\beta_2)^2_2$ denote the corresponding partition…

Let $\mathcal{D}(V,\kappa)$ denote the derived model at $\kappa$, let $\Theta^{D(V,\kappa)}$ be its supremum of ordinals surjected by its reals, and let $\kappa^+$ be the successor…

Let $V$ satisfy PFA, let $\kappa$ be a limit of Woodin cardinals, and let $D(V,\kappa)$ denote the derived model at $\kappa$. Let $\Theta^{D(V,\kappa)}$ be its Solovay ordinal. Wil…

Let $V$ satisfy PFA, let $\nobreak\bkappa$ be a limit of Woodin cardinals, and let $D(V,\kappa)$ denote the derived model at $\kappa$. Write $\Theta_0^{D(V,\kappa)}$ for the first…

PFA-to-superstrong-cardinal conjecture. Under these hypotheses, there is a superstrong cardinal in $\operatorname{HOD}^{L({{\sf{uB}}}_g,\mathbb{R}_g)}$.

Assume that there are unboundedly many Woodin cardinals and that the class of measurable cardinals is stationary. The principles $\sf{Sealing}$, $\sf{Sealing}^+$,…

Let $\force$ denote the Proper Forcing Axiom, let $\theta\text{-}\rm{UB}$ denote the relevant universally Baire framework, and let ${\mathbb{R}}_g$ and $\Gamma^\infty_g$ be the rea…

Assume $\mathrm{OCA}_\infty+\mathrm{MA}_{\aleph_1}$. Let $(M_i)$ and $(N_i)$ be sequences of separably representable type II$_1$ factors, and let $\bigoplus_\tau M_i$ be the ideal…

Let $X$ and $Y$ be locally compact Polish spaces. For a locally compact space $X$, write $X^* = \beta X\setminus X$ for its Stone–Čech remainder, and call a homeomorphism between s…

Let PFA denote the Proper Forcing Axiom, and let a supercompact cardinal be a cardinal with the usual supercompactness property. PFA equiconsistency conjecture. PFA is equiconsiste…

PFA conjecture. The following theories are equiconsistent:

Let $\Gamma_{\aleph_3}$ be a class of partial orders, and let maximality with respect to a theory mean the five conditions given in the preceding definition: inclusion of the relev…

Let $V$ be a model of ${\text{{\sf MM}}}^{++}$ together with large cardinals, and let $P\in V$ preserve ${\text{{\sf MM}}}^{++}$. Martin's maximum conjecture. For every such $P$, ……

PFA(S)[S] conjecture. Under $\mathrm{PFA}(S)[S]$, every first-countable perfect pre-image of $\omega_1$ includes a copy of $\omega_1$.