102 problems

Let $V$ be the universe of sets, let $Ord$ denote the class of ordinals, and let $L(V_{\alpha})$ be the constructible universe relativized to $V_{\alpha}$. For each $\alpha\in Ord$…

For each $n\in\omega$, let $b_{2n+1,0}$ denote the cardinal invariant defined in the source from the direct-limit images of the relevant iterates of $M_{2n+1}(x)$, and let…

Let $n\in\omega$. A cardinal is $\bf{\Gamma}$-reachable if there is a sequence of distinct $\bf{\Gamma}$-sets of that length, and is $\bf{\Gamma}$-unreachable otherwise. Kechris pr…

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.…

Let $\gamma$ be a limit ordinal, and let $(\mathrm{M}^{\alpha})_{\alpha\in\mathrm{Ord}}$ denote the sequence of inner mantles, with $\mathrm{M}^{\alpha}$ its stage at index…

Assume the hypotheses and notation of the paper's comparison theorem: $\mathcal{M}$ and $\mathcal{N}$ are countable premice, with the specified iterability, smallness, realizabilit…

Let $A_{(\alpha,n)}$ be the sets defined from $J_{\alpha}(\mathbb{R})$, and let a mouse set mean a set $A\subseteq\mathbb{R}$ of the form $A=\mathbb{R}\cap\mathcal{M}$ for some cou…

Let $c$ be the relevant set and let $X$ be the class or structure occurring in the preceding discussion. Say that a subset is bounded in $c$ when it is bounded with respect to the…

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 $M$ 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 ${\sf BMM}$ 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…

Weakening conjecture. The assertion $(\clubsuit)$ still holds under much weaker assumptions than the non-existence of $0^\P$.

Steel's conjecture. The failure of … boxvoidkappa $$ is at least as strong as the existence of a superstrong cardinal.

Let $({\mathcal{P}},\Psi)$ be a hod pair such that … Let $\kappa$ be the least strong cardinal which reflects the class of strong cardinals, and let…

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 $C^2(\alpha9omega)$ denote the inner model obtained from second-order definability over $\alpha9omega$, let $\text{CH}$ denote the continuum hypothesis, and let $M_n$ denote th…

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…

The Extender Embedding Axiom (EEA) is the assertion concerning elementary embeddings associated with the relationship between superstrong and tall cardinals described in the surrou…

Rudominer's conjecture. If $[\beta,\beta]$ is a projective-like $\mathcal{S}$-gap which is not the successor of a strong $\mathcal{S}$-gap, then…

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