Vaught's Conjecture

• Morley proved that for a first-order theory $T$ in a countable language, either $I(T, \aleph_0) \leq \aleph_0$, $I(T, \aleph_0) = \aleph_1$, or $I(T, \aleph_0) = 2^{\aleph_0}$. In other words, any counterexample to Vaught's Conjecture must have exactly $\aleph_1$ countable models up to isomorphism.
• The conjecture is known to hold for a number of special classes of theories, including:
  • $\omega$-stable theories (Harrington, Makkai and Shelah).
  • o-minimal theories (Mayer).
  • Superstable theories of finite U-rank (Buechler).
  • Theories of trees—i.e. theories in a language with one binary relation symbol $<$ which extend the theory stating that $<$ is a partial order and that the set of predecessors of each element is linearly ordered (Steel).

Vaught's Conjecture is among the oldest and most famous open problems in mathematical logic and has close connections to model theory, descriptive set theory and computability theory. It was first posed by Vaught in 1961 [Vau61].

There are many well-known strengthenings of Vaught's Conjecture, including the topological Vaught's Conjecture, Martin's Conjecture (the model theoretic version), and the $\omega$-Vaught's Conjecture.