Vaught Conjecture

In 1961 Vaught asked how many countable models, up to isomorphism, a complete theory $T$ in a countable first-order language can have [Vaught1961]. The conjecture asserts that this number $I(T,\aleph_0)$ is either at most countable or exactly $2^{\aleph_0}$; equivalently, when $2^{\aleph_0}>\aleph_1$, the intermediate value $\aleph_1$ never occurs. It can be read as a definability-flavored counterpart of the continuum problem for the isomorphism relation on countable models.

Morley proved that $I(T,\aleph_0)$ is always either at most $\aleph_1$ or exactly $2^{\aleph_0}$, so the conjecture amounts precisely to ruling out the value $\aleph_1$ [Morley1970]. It has been verified for many major classes of theories, among them the $\omega$-stable theories, and Steel established further cases, alongside descriptive-set-theoretic analyses of the problem [Steel1978Vaught].

A resolution requires either a structure theory strong enough to exclude exactly $\aleph_1$ countable models in general, or a counterexample, which none of the known classes provides; for an arbitrary complete theory in a countable language the conjecture remains open.