365 problems

In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail…

In mathematics, specifically set theory, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:There is no set whose cardinality is strictly…

Chang's Conjecture.

Hurewicz conjecture. A set of reals has the Hurewicz property $\mathsf{U}_{\mathrm{fin}}(\mathcal{O},\Gamma)$ if and only if it is a countable union of its compact subsets.

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

Let $x$ be an infinite set, let $\omega$ be the first infinite ordinal, and let $\mathcal P(\omega)$ denote the power set of $\omega$. Cantor's continuum hypothesis. There is no ca…

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…

Shelah's Strong Hypothesis. For every singular cardinal $\sigma$,

Let $\mathcal{L}_{\omega_1,\omega}$ be the infinitary logic allowing countable conjunctions and disjunctions but only finite strings of quantifiers. Write $\aleph_{\omega_1}$ for t…

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 ${\mathcal H}=l^2(\omega)$ be a separable infinite-dimensional Hilbert space, let ${\mathcal B}$ be the algebra of bounded operators on ${\mathcal H}$, let…

Milner–Sauer conjecture. The poset $P$ must contain an antichain of size $\kappa$.

Mathias's question. In Solovay's model, are there infinite MAD families? More generally, is it a theorem of $\mathsf{ZF}+\mathsf{DC}_{\mathbb R}$ that there are no infinite MAD fam…

Let $2^\omega$ be the Cantor group under coordinatewise addition, and call $X\subseteq 2^\omega$ strongly meager if, for every measure zero set $H$, one has … Dual Borel Conjecture…

A subset $X\subseteq\mathbb{R}$ is full measure universal if every Lebesgue measurable subset of $\mathbb{R}$ whose complement has Lebesgue measure zero contains an affine copy of…

Let $\mathbb{G}$ be a Polish group, and consider the Galvin–Mycielski–Solovay property that strong measure zero sets are characterized by avoiding translates or products with every…

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…

$(2,3)$-selective ultrafilter conjecture. The existence of a $(2,3)$-selective ultrafilter implies the existence of a nowhere dense ultrafilter. Consequently, if there are no nowhe…

Let $(\kappa_1,\kappa_0)$ and $(\mu_1,\mu_0)$ be pairs of cardinals. For a structure $\mathfrak A$ on $\kappa_1$ in a countable language, write $\mathfrak B\prec\mathfrak A$ when…

Debs–Saint Raymond conjecture. For every analytic ideal $\mathcal{I}$ and $0<\alpha<\omega_1$,

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 $\mathsf{CH}$ denote the continuum hypothesis, and let $\mathsf{ZFC}$ denote Zermelo–Fraenkel set theory with the axiom of choice. A statement is independent of $\mathsf{ZFC}$…

Shelah's PCF conjecture.

Telgársky's conjecture. For every $k\geq 2$, there is a space $X_k$ such that player ${\scriptsize\operatorname{NONEMPTY}}$ has a winning $(k+1)$-tactic in the Banach–Mazur game on…

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…