73 problems

Woodin's HOD conjecture. … mathrm{ZFC}+text{“there is a supercompact cardinal”} $$ proves the HOD hypothesis.

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…

Steel's conjecture. Assume $AD$. Every $T$-invariant function is equivalent to a uniformly $T$-invariant function on a cone.

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

Let $\mathcal{V}$ be the relevant model and let $\delta$, $\mathcal{Q}$, and $\eta$ be as in the preceding results. The cardinal $\delta^{\mathcal{Q},\eta}_{\infty}$ is considered…

Under $AD^+$, let an ordinal definable real mean a real definable from ordinal parameters, and let a real appear in an $d714_1$-iterable mouse mean that it belongs to such a mouse.…

For each formula $\phi(x_1,\ldots,x_n)$ and stationary set $S\subseteq\omega_1$, let $Y_{\phi}$ be the set of all $a\in\{0,1\}^{\omega_1}$ such that there is a stationary…

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…

Consistency-strength conjecture. The theory

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…

Failure-of-square equiconsistency conjecture. The following theories are equiconsistent:

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

Let $V$ be a model of … For a set of reals $A$ and a set-generic extension $V[G*H]$, write $A^{V[G]}$ and $A^{V[G*H]}$ for the interpretations of $A$ in the relevant extensions. Ge…

A pointclass of universally Baire sets of reals is called productive when it is closed under complements and projections, and its projections are absolute between set-generic exten…

Sargsyan's conjecture. Conclusion (2) of that conjecture should hold: the model $N$ has

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

Upper-bound conjecture. A pure extender mouse with proper classes of Woodin cardinals, strong cardinals, and strong cardinals reflecting strong cardinals is a consistency strength…

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

Amenable powerset conjecture. We conjecture that ${{\sf{uBp}}}$ is the amenable powerset of ${{\sf{uB}}}$.

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

Minimal model conjecture. Suppose there is a ${\sf{HOD}}$-Berkeley cardinal. Then the minimal model of

Determinacy classification conjecture. Under $\mathsf{Det}(\mathbf{\Sigma}(\omega^\omega))$, Theorem van Engelen holds for all zero-dimensional $\mathbf{\Sigma}$ spaces that are no…

Assume $\mathsf{AD}^{L(\mathbb R)}$, and let $\mathcal{M}$ be an inner model operator in $L(\mathbb R)$. Steel's conjecture. For a cone of reals $x$, there is a wellorder 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…