33 problems

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…

The theories in question are ${\sf PFA}$ and ${\sf ZFC}+\exists$ a supercompact cardinal. Equiconsistency conjecture. These theories are equiconsistent. This is presented as a conj…

Let $W\subseteq V$ be models of Martin's Maximum, denoted by ${\sf MM}$, with the same cardinals. For an ordinal set $A$, write $[\operatorname{Ord}]^{\leq\omega_1}$ for the collec…

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 $M$ be an inner model of $V$. Write $2^{\omega_1} \cap M$ for the subsets of $\omega_1$ belonging to $M$, and write $\operatorname{Ord}^{\omega_1} \cap M$ for the functions fro…

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 ${\sf BMM}$ denote the Bounded Martin's Maximum forcing axiom, and let an inner model with a Woodin cardinal mean an inner model containing a Woodin cardinal. BMM inner model c…

Let \mathop{\raisebox{3pt}{\framebox[6pt]{}}}{\setbox1=\text{text{sc mp}}\baselineskip=0pt\vtop{\text{text{sc mp}}\text to\wd1{\hfilsim\hfil}}}{} denote the maximality prin…

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 $\mathsf{BA}$ denote the assertion that for all $\aleph_1$-dense subsets $A,B\subseteq\mathbb R$ there is an autohomeomorphism of $\mathbb R$ mapping $A$ to $B$. For a topologi…

Let $T$ be a theory and let $\varphi$ be a statement in the language of set theory. Write $T \models_\Omega \varphi$ when $\varphi$ is true in every rank-initial segment of every s…

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 $\operatorname{HOD}^{L({{\sf{uB}}}_g,\mathbb{R}_g)}$.

OCA$_T$ rigidity conjecture. $\operatorname{{\mathsf {OCA}_T}}$ alone suffices for all rigidity results for quotients of reduced products of countable structures modulo analytic id…

OCA$_T$ conjecture. The conclusion of Proposition $$ follows from $\operatorname{{\mathsf {OCA}_T}}$ alone.

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…

Assume $\mathrm{OCA}_\infty+\mathrm{MA}_{\aleph_1}$. Let $A$ and $B$ be unital, separable $\C^*$-algebras, and let $\mathcal K(H)$ denote the compact operators on $H$. Stabilized c…

Let $A$ and $B$ be separable $\C^*$-algebras, and let a corona isomorphism be called trivial when its graph is Borel in the product of the strict topologies on the multiplier algeb…

Let $A$ be a separable, nonunital $\C^*$-algebra, and let $\mathscr{Q}(A)$ denote its corona algebra. An automorphism of $\mathscr{Q}(A)$ is trivial when its graph is Borel in the…

Let $\textup{MM}(S)[S]$ denote the stated forcing axiom, and let SRP (the Strong Reflection Principle) be the principle asserting that whenever $[?]$ [the principle is defined in t…

Proper Forcing Axiom equiconsistency conjecture. $PFA$ is equiconsistent with the assertion that there exists a supercompact cardinal:

Let $\delta$ be an ordinal, and let $\mathbb{T}^{\omega_2}_\delta$ denote the tower of normal filters of height $\delta$ concentrating on $P_{\omega_2}$. A tower is strongly presat…