101 problems

Shelah's conjecture. It is consistent that every Aronszajn line contains a Countryman suborder.

Let $X$ be an ordinal-definable set of reals, and let $\mathbb{P}$ denote Jensen's “minimal $\Pi^1_2$ real singleton forcing”. Let $\mathbb{P}^{\omega}_{\mathrm{fs}}$ be the finite…

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 $\alpha\in Ord$…

Let $V$ be a model of $\mathsf{ZF}+\mathsf{KWP}$, let $G$ be a $V$-generic filter, and let $W$ be an intermediate model of $\mathsf{ZF}$ with … Here, $\mathsf{KWP}$ means that ever…

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 $V$ be a model of set theory with a large cardinal, for example an extendible cardinal. Woodin's Axiom of Choice conjecture. If $V$ has a large cardinal, e.g., an extendible ca…

Assume that $V$ satisfies $\mathsf{ZF}$ and that $\delta$ is an extendible cardinal. Let $G$ be $V$-generic for a forcing that collapses $V_\delta$ to be countable. Axiom of Choice…

Let $\mathbb{S}$ be a Suslin ccc poset. A Suslin ccc poset is correctness-preserving if, given a correct diagram $\langle\mathbb{P}_0,\mathbb{P}_1,\mathbb{Q}_0,\mathbb{Q}_1\rangle$…

Consistency-strength dichotomy conjecture. The consistency strength of this statement is always either ZFC or ZFC together with the existence of an inaccessible cardinal; moreover,…

Villaveces's conjecture. The consistency strength of the failure of the GCH at an unfoldable cardinal is no greater than that of a strongly unfoldable cardinal.

Assume that $\lambda_n$ is the $n$-th inaccessible cardinal, let $\kappa=\sum_{n<\omega}\lambda_n$, and consider a forcing notion $\mathbb P$. Shelah's forcing conjecture for pseud…

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…

Consistency conjecture.

Countable-support iteration conjecture. The conclusion of Theorem should hold for the countable support iteration of the partial order ${\Bbb P}({\cal I})$.

Consistency conjecture. It is consistent that every non-principal ultrafilter on $c9$ is not nowhere dense and, simultaneously, is not a $J_\alpha$-ultrafilter for any $b1<c9_1$.

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 $

Cover-removal conjecture. The added assumption that $V\subseteq\overline{V}$ satisfies the $\kappa$ cover property is unnecessary: the $\delta$ approximation and cover properties a…

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…

The paper discusses definable ideals $I$ whose associated forcing $P_I$ is proper but may fail to be $<\omega_1$-proper, or whose properness may fail to be absolute throughout forc…

Let $\mathbb B$ be a complete Boolean algebra, and let $\mathcal G^\textsf{\textup{c\&c}}_{<\aleph_{\omega+1}}(\mathbb B)$ denote the cut-and-choose game of length less than…

Let $\operatorname{\mathscr{L}}$ be a language, let $X$ be a set, and let $\kappa$ be the relevant cardinal. Let $\AgeK$ be a strong Fraïssé class in $\operatorname{\mathscr{L}}$,…

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…