Vaught conjecture

Remarks

The conjecture is open. Morley proved in 1970 that $I(T, \aleph_0)$ is at most $\aleph_1$ or equal to $2^{\aleph_0}$. So the conjecture holds under the continuum hypothesis, and the open question is whether a theory can have exactly $\aleph_1$ countable models when $\aleph_1 < 2^{\aleph_0}$. Shelah, Harrington and Makkai proved the conjecture for $\omega$-stable theories. Vaught himself proved that $I(T, \aleph_0)$ is never equal to $2$.

All the allowed values occur. The theory of dense linear orders without endpoints has exactly one countable model. The theory of algebraically closed fields of characteristic $0$ has $\aleph_0$ countable models, one for each transcendence degree $0, 1, 2, \dots, \aleph_0$. The complete theory of $(\Z, <)$ has $2^{\aleph_0}$ countable models.