39 problems

Let $\Theta^\infty=\Theta^{L(\Gamma^\infty,\mathbb{R})}$ and let $L(\mathcal{A}^\infty)$ be the model generated by $\mathcal{A}^\infty$. Square-sequence conjecture. … The claim is…

Martin's conjecture. Under these assumptions: (I) if $f$ is degree-invariant and is not increasing a.e., then $f$ is constant a.e.; and (II) $\leq_m$ pre-well-orders the set of deg…

Martin's conjecture. Assume $AD$. Then:

Assume ${\sf{AD}}^+$. Let $\kappa$ be a regular Suslin cardinal, and let ${\mathbf{\Gamma}}$ be the pointclass of $\kappa$-Suslin sets. Suslin-cardinal unreachable successor conjec…

Let ${\mathbf{\delta}}^1_{2n+2}$ denote the supremum of the lengths of ${\mathbf{\Delta}}^1_{2n+2}$ pre-well-orderings of ${\mathbb{R}}$. Assume ${\sf{ZF+AD+DC}}$. Kechris's second…

Let ${\mathbf{\delta}}^1_{2n+2}$ denote the supremum of the lengths of ${\mathbf{\Delta}}^1_{2n+2}$ pre-well-orderings of ${\mathbb{R}}$. Assume ${\sf{ZF+AD+DC}}$. Kechris's first…

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 ${\mathsf{AD}}$ denote the axiom of determinacy and let ${\mathsf{Bl}\text{-}\mathsf{AD}}$ denote the axiom of Blackwell determinacy. Martin's conjecture.…

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

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

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

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…

Let $V$ satisfy $obreak\textsf{AD}_{\mathbb R}+ obreak\textsf{NLE}+V=L({\wp}(\mathbb R))$. Let $obreak\Delta\subseteq{\wp}(\mathbb R)$ and let…

Let $obreak\textsf{AD}_{\mathbb R}$ denote determinacy for real games. Suppose that for every set $B\subseteq\mathbb R$ there is a function $F$ as in Theorem nx such that $F$ is n…

Let $obreak\textsf{AD}^+$ denote the strengthened axiom of determinacy. For a function $F$ as in the paper's Theorem nx, call $F$ pe-bad when, for a cone of reals $x$, the supremu…

Let $obreak\textsf{AD}^+$ denote the strengthened axiom of determinacy, and let $obreak\textsf{NLE}$ denote the no-long-extender assertion. Let $obreak\textsf{HPC}$ denote hod-p…

Let $obreak\textsf{AD}^+$ denote the strengthened axiom of determinacy, and let $obreak\textsf{NLE}$ be the no-long-extender assertion. Let $obreak\textsf{LEC}$ denote…

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…

Suppose $\kappa$ is a supercompact cardinal and there is a proper class of inaccessible limits of Woodin cardinals. Let $g\subseteq\operatorname{Col}(\omega,2^{2^\kappa})$ be $V$-g…

The principles $\mathsf{GBT+DC}$ and $\mathsf{ZF}$ are considered as set-theoretic theories, where $\mathsf{GBT}$ denotes Baire's grand theorem and $\mathsf{DC}$ denotes the axiom…

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

Assume $AD^+$. Let $\kappa$ be a regular Suslin cardinal. Sargsyan's conjecture. The successor cardinal $\kappa^+$ is $S(\kappa)$-unreachable. The abstract presents this as the ana…

Let $LSA$ abbreviate the assertion that $AD^+$ holds and there is a largest Suslin cardinal belonging to the Solovay sequence. Let $\vec{E}$ be an extender sequence and write…