63 problems

An ordinal definable game of length $\omega_1$ on natural numbers with real parameters is a game in which the payoff and defining data are ordinal definable, with real parameters a…

Let $G_{\operatorname{cont}}(\nu,A)$ be the game of continuously coded length, where $\nu$ is a partial function from reals to natural numbers and $A$ is a payoff set of sequences…

Equiconsistency conjecture. The existence of an indestructible weakly compact cardinal is equiconsistent over $ZFC$ with the existence of a supercompact cardinal.

Consistency-strength claim. The consistency strength of the theory $

Let $C(\mathop{\text{aa}})$ and $C^2(\omega)$ denote the inner models defined in the paper, with $C(\mathop{\text{aa}})$ the model generated using the relevant class of structures…

Let $C^2(\alpha9\omega)$ denote the inner model obtained from second-order definability over $\alpha9\omega$, let $\text{CH}$ denote the continuum hypothesis, and let $M_n$ denote th…

Proof-theoretic dilator conjecture. The proof-theoretic ordinals satisfy

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

Chang-type uB-powerset sealing conjecture. Moreover, the $\sf{Sealing}$ Theorem holds for $\mathsf{C-uBp}$ in $V[g]$.

Ultraexacting-above-extendible rank-Berkeley model conjecture. There is a set model of ZF with a rank-Berkeley cardinal.

Let $\mathrm{EEA}$ denote the Extender Embedding Axiom, and let $V=\mathrm{Ultimate-}L$ denote the corresponding Ultimate-$L$ axiom. Woodin's Ultimate-$L$ conjecture. The theory ……

Solovay-sequence conjecture. Under these assumptions, $V_\zeta^{{\sf{HOD}}_X}$ is a model of

Least-cardinal structural reflection conjecture. A cardinal $\kappa$ is the least cardinal satisfying some large-cardinal notion if and only if $\kappa$ is the least cardinal satisfy…

Structural reflection characterization conjecture. Every known large cardinal notion is equivalent to some form of SRP.

Let $(\mathcal{V},\Omega)$ be an excellent least branch hod pair with $\mathcal{V}\models\mathsf{ZFC}$, let $\delta$ be a Woodin limit of Woodin cardinals in $\mathcal{V}$, and let…

Let $\delta$ be a Woodin cardinal that is a limit of Woodin cardinals. Let $g\text{?}$ be $V$-generic for $\mathrm{Col}(\omega,{<}\delta)$, and let $L(\Gamma^*_g,\mathbb{R}^*_g)$ b…

Covering with Chang Models conjecture. The asserted transitive model $M$ exists with all the listed properties. This conjecture is intended to overcome limitations of the core mode…

Let $\mathfrak{Sp}^*$ denote the spectrum introduced in the surrounding discussion, and let a cardinal $\kappa$ be virtually strong if for every $\gamma \geq \kappa$ and every…

Let \textit{\th}_0=\omega, let \textit{\th}_{\alpha+1}=\textup{\TH}(\textit{\th}_\alpha^\omega), and for limit $\gamma$ let…

Assume $AD^+$ and that there is a largest Suslin cardinal which is a member of the Solovay sequence. A transitive model $M$ of $ZF$ is a minimal model of $\Theta strong$ when…

Assume ${\sf{NLE}}$ and suppose there are unboundedly many Woodin cardinals and strong cardinals. Let $\kappa$ be a limit of Woodin cardinals and strong cardinals such that either…

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…