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Vaught's theorem establishes an upper bound on the number of countable models of a complete theory: the number is either 1 (categorical) or ≥ ℵ₀. There is no complete theory with exactly 2 countable models. This surprising rigidity reflects the discreteness of first-order logic and is a key result in counting models.
From your study of countable models, you know that a complete first-order theory always has at least one countable model (by the downward Löwenheim-Skolem theorem). From the spectrum of a theory, you may have studied how many non-isomorphic models a theory can have at a given cardinality. Vaught's theorem is a striking constraint on this count at the countable level: the number of non-isomorphic countable models of a complete theory can never equal exactly 2.
The result is counterintuitive. You might expect that by tuning a theory's axioms you could produce exactly 2 distinct countable structures. Vaught's theorem says no: the count is either 1 (the theory is ℵ₀-categorical, all countable models are isomorphic to each other), or it is at least ℵ₀. The argument proceeds through types — maximal consistent sets of formulas in one free variable that describe the possible "behavior" of a single element in a model. Two countable models are non-isomorphic exactly when they realize different collections of types. The key insight is that if a type is not isolated (not implied by a single formula of the theory), then omitting or realizing it generates further choices, each spawning more non-isomorphic models — and this cascade cannot stop at exactly 2.
To see the intuition more concretely: suppose a theory has two non-isomorphic countable models M and N. The difference between them is witnessed by some non-isolated type p that is realized in one but not the other. But the theory's combinatorial structure means there are infinitely many "variants" of p — partial types extending it in incompatible directions, each realizable in some countable model. The Omitting Types Theorem guarantees that any non-isolated type can be omitted in a countable model; conversely, isolated types must be realized. The interaction between isolated and non-isolated types therefore generates infinitely many distinct realizations, ruling out a count of exactly 2.
Vaught's theorem motivates Vaught's conjecture, one of the most important open problems in model theory: must the number of countable models of a complete theory be either at most ℵ₀ or exactly 2^ℵ₀? (Under the continuum hypothesis these are the only options anyway; the question is non-trivial when CH fails.) The conjecture has been proved for special classes of theories (ω-stable theories, theories without the independence property) but remains open in general. Vaught's theorem is thus the opening move in a deep classification project for first-order structures, revealing that the spectrum of countable models obeys unexpectedly rigid constraints.