38 problems

Let $\mathfrak s$ and $\mathfrak r$ denote the splitting and reaping numbers, and let $\mathfrak c$ denote the continuum. Consider models obtained by the shattered iteration of eit…

Let $\mathfrak b$, $\mathfrak d$, and $\mathfrak c$ denote the bounding number, dominating number, and continuum, respectively. Assume … and let $\kappa,\lambda$ be regular cardina…

Correctness-preservation conjecture. The diagram $\bar{\mathbb E}$ is correct.

Let $V$ be the ground model. For a centreless group $G\in V$, let $\tau(G)$ be the least ordinal $\alpha$ such that the automorphism tower satisfies $G_\beta=G_\alpha$ for all…

Let $IFS$ be the axiom schema asserting that for every weakly absolutely definable set $X$ and every partial order $P$ definable in $L(X)$, if … then there is an $L(X)$-generic sub…

Let $I$ be a $c3$-ideal on $2^\omega$ or $\omega^\omega$, and let $P_I$ be the corresponding factor poset. Assume that $P_I$ is proper and has continuous reading of names. Trace-id…

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

A MAD family is a maximal almost disjoint family of infinite subsets of $\text{omega}$. A family is concentrated on a countable subset of itself if there is a countable subfamily s…

A MAD family is a maximal almost disjoint family of infinite subsets of $\text{omega}$, and a subset $X\subseteq 2^\omega$ is a sigma-set if for every Borel set…

Let $X$ be a set, let $\operatorname{\mathscr{L}}$ be a language, and let $\kappa$ be the relevant cardinal. Let $\mu_0$ and $\mu_1$ be invariant measures on…

Let $X$ be the underlying set and let $\AgeK_0$ and $\AgeK_1$ be non-trivial Fraïssé classes, each with at most $|X|$ many elements up to isomorphism. Write $\CohenX$ for the Cohen…

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 $T$ be a first-order countable unstable theory. Two models $M$ and $N$ of $T$ are random twins if they are non-isomorphic but become isomorphic after the universe is extended b…

Let $L$ be the constructible universe, and let $\theta$ range over the reflecting ordinals $\sigma^1_n$ and $\pi^1_n$. Consider forcing over $L$ with countable-support iterations o…

Let $n\geq 2$. Let $\mathscr{L}_n=\{R\}$ be a relational language with $R$ an $n$-ary relation, and let $\operatorname{AgeK}_{\mathrm{sym},n}$ be the class of all finite…

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…

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…

Let $\mathrm{L}$ be a relational $\omega$-signature. A theory in $\mathrm{L}_{\infty\infty}$ is Boolean satisfiable if it has a Boolean-valued model satisfying the theory. A Boolea…

Let $V=L$. Let $\mathbb{P}$ be a ccc forcing notion and let $G\subseteq\mathbb{P}$ be generic. Write $\clubsuit_S$ for the clubsuit principle associated with a stationary set…

SOCA($\aleph_2$) consistency conjecture. The statement $\textbf{SOCA}(\aleph_2)$ is consistent.

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…

Let $d702$ 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 $\alpha$.…

Let $G\subseteq \mathbb{M}[\vec{U}]$ be a $V$-generic filter, where … Let $V\subseteq M\subseteq V[G]$ be a transitive $ZFC$ model. Intermediate-model classification conjecture. Ei…

Let $V$ be a model satisfying the Kinna–Wagner principle $\mathsf{KWP}$, let $G$ be a $V$-generic filter, and let $M$ be an intermediate model with $V\subseteq M\subseteq V[G]$. Ki…

Let $\text{SCC}^{\text{cof}}$ and $\text{SCC}^{\text{split}}$ denote the two set-theoretic principles introduced in the surrounding discussion. Non-reversibility conjecture. The im…