Problem #1

Vaught's Conjecture

Open!!!

Let $T$ be a complete first-order theory in a countable language, and let $I(T, \aleph_0)$ denote the number of countable models of $T$ up to isomorphism. Vaught's Conjecture states that

I(T, \aleph_0) \le \aleph_0 \quad \text{or} \quad I(T, \aleph_0) = 2^{\aleph_0}.

In other words, a complete countable theory cannot have a number of countable models that is strictly between countable and continuum many.

Model Theory

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Reference for the problem statement

Robert L. Vaught, Denumerable models of complete theories, Infinitistic Methods (Proc. Symposium on Foundations of Mathematics, Warsaw), 1961

Definitions

Complete theory: a theory $T$ such that for every sentence $\varphi$ in its language, either $T \vdash \varphi$ or $T \vdash \neg\varphi$ .
$I(T, \aleph_0)$ : the number of pairwise non-isomorphic countable models of $T$ .

Known Partial Results

The conjecture is known to hold for theories that are $\omega$ -stable (Shelah, Harrington, Makkai).
It is also known for theories of trees and for various other restricted classes.
By a theorem of Morley, if $T$ has uncountably many countable models, it has either exactly $\aleph_1$ or exactly $2^{\aleph_0}$ many — so under CH the conjecture is trivial, but the independent case remains genuinely open.

Notes

Vaught's Conjecture is widely regarded as one of the central open problems in model theory, with deep connections to descriptive set theory (via the Topological Vaught Conjecture) and to the theory of $\Pi^1_1$ equivalence relations.

Additional References

Vaught, R. L. "Denumerable models of complete theories." Infinitistic Methods, 1961.
Shelah, S. "Classification Theory." North-Holland, 1978 (revised edition 1990).
Steel, J. "On Vaught's conjecture." Cabal Seminar 76–77, Lecture Notes in Mathematics, 1978.

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