29 problems

An Abstract Elementary Class is a class of structures equipped with a strong substructure relation; $\text{LS}(d53e)$ denotes its Löwenheim–Skolem number, and $d53e$ is categorical…

Let $T$ be a complete theory with quantifier elimination in a relational language $L$, and let $P$ be a distinguished unary predicate symbol in $L$. For a model $M\models T$, write…

Assume that $T$ is a small first-order theory. Let $L_1(T)$ be the smallest countable fragment of $L_{\omega_1,\omega}$ containing all first-order formulae and, for every…

Łoś's conjecture. $Cat(T)$ contains every uncountable cardinal or no uncountable cardinal.

Łoś's categoricity conjecture. If $T$ is categorical in some uncountable cardinality, then $T$ is categorical in every uncountable cardinality.

Let $\operatorname{\mathcal{K}}$ be an abstract elementary class, let $\operatorname{Hanf}(\operatorname{\mathcal{K}})$ and $LS(\operatorname{\mathcal{K}})$ denote its Hanf and Löw…

Let $\operatorname{\mathcal{K}}$ be an abstract elementary class, let $\operatorname{Hanf}(\operatorname{\mathcal{K}})$ be its Hanf number, and let $LS(\operatorname{\mathcal{K}})$…

Let $\operatorname{\mathcal{K}}$ be an abstract elementary class. Say that $\operatorname{\mathcal{K}}$ is categorical in a cardinal when it has exactly one model of that cardinali…

A theory is countably categorical when it has, up to isomorphism, exactly one countable model, and it is stable when it has the model-theoretic stability property. Superstability c…

Let $T$ be a complete theory with quantifier elimination in a relational language $L$, and let $P$ be a distinguished unary predicate symbol in $L$. For a model $M\models T$, write…

Stable aleph-null-categorical superstable conjecture. Every stable $\aleph_0$-categorical theory is superstable.

Zilber's conjecture. Every non-locally modular geometry of a strongly minimal set behaves like a field.

Let $T$ be a countable complete first-order theory with a distinguished predicate $P$. Say that $T$ is categorical over $P$ when it has at most one model, up to isomorphism over…

Let $T$ be a first-order theory. Los–Morley categoricity conjecture. The supplied excerpt begins the statement but does not provide the assertion following the setup. The conjectur…

Let ${\bf K}$ be an abstract elementary class and let $\lambda=\operatorname{LS}({\bf K})$, its Löwenheim–Skolem number. Shelah's Hanf-number threshold conjecture. The threshold fo…

Let $\mathcal{T}$ be the class of $\aleph_0$-categorical, $\aleph_0$-stable classical theories in countable languages, and let $\mathcal{D}\subseteq\mathcal{T}$ consist of those th…

Let $Q$ be an $\aleph_0$-categorical, $\aleph_0$-stable theory in a countable language, and let $Q_P$ denote the theory of beautiful pairs of models of $Q$. Approximate categoricit…

Grossberg's conjecture. If

Let $\mathcal{A}$ be a nice accessible category with directed colimits having just one model of presentability rank $\kappa$. Weaker conjecture. Then $\mathcal{A}$ must be categori…

Naive categoricity conjecture for Betti cohomology. The description of a Weil cohomology theory with $\mathbb Q$-coefficients that factors through $\operatorname{Mot}/K$ is categor…

Eventual categoricity conjecture. There is a threshold cardinal $ctheta$ such that if $cpsi$ is categorical in some $cmu cge ctheta$, then $cpsi$ is categorical in all…

Strong categoricity conjecture. If $cpsi$ is categorical in some $cmu cge cbeth_{comega_1}$, then $cpsi$ is categorical in all $cmu' cge cbeth_{comega_1}$.

Let ${\mathfrak k}$ be an abstract elementary class and put $\kappa={\rm LST}_{\mathfrak k}+|\tau_{\mathfrak k}|$. Superlimit-or-nonstructure conjecture. At least one of the follow…

Stationarity dichotomy conjecture. The classes $S_1$ and $S_2$ are not both stationary. A weaker version replaces $S_2$ by the class of $\lambda\in C^{\rm fp}_{\aleph_0}$ such that…

Let ${\mathfrak k}=(K_{\mathfrak k},\leq_{\mathfrak k})$ be an abstract elementary class, with Löwenheim–Skolem–Tarski number ${\rm LST}_{\mathfrak k}$. Categoricity transfer conje…