Vaught's conjecture

Can a countable first-order theory have strictly between ℵ₀ and 2^ℵ₀ countable models?

MSC 03 · Mathematical logic and foundations

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

Conjecture: for any complete first-order theory $T$ in a countable language, the number of countable models of $T$ up to isomorphism is either at most $\aleph_0$ or exactly $2^{\aleph_0}$ .

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Background

Posed by Robert Vaught in 1961. Under the continuum hypothesis the statement is trivially true, so the content is entirely in models of set theory where CH fails; this makes it a rare model-theoretic problem whose difficulty is not resolved by adding axioms.

Known partial results

Morley (1970) proved the number of countable models is at most $\aleph_1$ or exactly $2^{\aleph_0}$ , reducing the problem to excluding exactly $\aleph_1$ .
Shelah, Harrington and Makkai proved the conjecture for $\omega$ -stable theories.
The topological Vaught conjecture, for Polish group actions, remains open and is at least as hard.

Recorded progress

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