22 problems

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…

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…

Cofinality conjecture. For $\kappa$ a characterizable cardinal, $\kappa$ is not homogeneously characterizable if and only if

Natural complete axiom-system conjecture. There is a natural complete axiom system for all $L^{St}(S_0,S_1,...)$ based on the usual axioms and rules of $\mathcal{L}(Q_1)$.

Let $m\geq 1$. Write $\mathcal{A}_{m+1}$ for the canonical tree of rank $m+1$, and let $B^{(m)}_j:=[m+1:j]$ be the tree obtained by deleting all but $j$ of the level-one subtrees o…

Let … -theory, with Vaught ordinal denoted by $\operatorname{vo}(T)$. The $\omega$-Vaught conjecture. Every theory has a countable Vaught ordinal, and every…

Let $k$ be a positive integer, and let $!^m\nabla \mathrm{ACT}_\omega^{(k)}$ denote the $k$-th fragment of infinitary action logic with multiplexing. Its derivability problem is me…

Lambda-omitting types conjecture. The $\lambda$-omitting types theorem also provides a characterization of the maximality of $\mathcal{L}_{\omega\omega}^{-}$.

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 $S$ be a consistency property for $\mathrm{L}_{\infty\infty}$. Mansfield's completeness theorem produces a Boolean-valued model $\mathcal{M}_S$ over…

Successor-stage independence conjecture. It is independent of the axioms of $\mathrm{ZFC}$ whether the Scott sentence of the $(\mathcal{M},\mathcal{N})$-full model characterizes…

Characterizing-sentence independence conjecture. It is independent of the axioms of $\mathrm{ZFC}$ which sentence in the construction has this desired characterizing property.

Closure-ordinal conjecture. The closure ordinal for this system is less than or equal to $\omega^\omega$.

Noninterpretability conjecture. There do not exist $L_{\omega_1\omega}$ formulas that, for all directed graphs $G$, define an interpretation of $G$ in $\Phi(G)$.

Let a PC class or PC' class be a class of structures axiomatized, respectively, by a pseudo-elementary theory or by a single pseudo-elementary sentence. Let…

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 $\varphi$ be an $L_{\omega_1\omega}$-sentence. It is scattered when it has no perfect set of pairwise nonisomorphic countable models. Absolute Vaught conjecture. If $\varphi$ i…

Let $K$ be a class of finite structures, let $t(n)$ be polynomially bounded, and let $r(s, mathcal{A})$ denote the relevant iteration-depth parameter for a structure $mathcal{A}$ a…

Let $phi$ be an $\mathcal{L}_{omega_{1},omega}$-sentence of quantifier depth $alpha$, and let Scott rank refer to the Scott rank of a model of $phi$. Uniform Vaught's conjecture. T…

Let an intermediate logic mean a logic stronger than ${\mathbb L}_{\lambda^+,\aleph_0}$ but weaker than ${\mathbb L}_{\lambda,\lambda}$, with the stronger demand that Ehrenfeucht–M…

Let $\Sigma$ be a signature, and let $\Sigma$-structures in a Grothendieck topos $\mathcal E$ be the corresponding category of models. Moerdijk's conjecture concerns the result cha…