4 problems

Let $T$ be a first-order theory, and consider its countable models up to isomorphism. Vaught's conjecture. The number of countable isomorphism types of models of $T$ is either coun…

Computable Scott sentence conjecture. For each even $n\geq 2$, there is a computable structure $\mathcal{M}$ with a $\Pi_n$ Scott sentence but no computable $\Sigma_{2n}$ Scott sen…

For a theory $T$ and ordinals $\alpha\geq\beta$, say that a $\Pinf{\alpha}$ theory is non-splitting when it has exactly one $\Pinf{\alpha+1}$ theory as a descendant. In the case of…

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