Vaught Conjecture - Vaught's Theorem

Vaught's Theorem

Vaught proved that the number of countable models of a complete theory cannot be 2. It can be any finite number other than 2, for example:

• Any complete theory with a finite model has no countable models.
• The theories with just one countable model are the ω-categorical theories. There are many examples of these, such as the theory of an infinite set.
• Ehrenfeucht gave the following example of a theory with 3 countable models: the language has a relation ≥ and a countable number of constants c₀, c₁, ...with axioms stating that ≥ is a dense unbounded total order, and c₀< c₁<c₂... The three models differ according to whether this sequence is unbounded, or converges, or is bounded but does not converge.
• Ehrenfeucht's example can be modified to give a theory with any finite number n≥3 of model by adding n−2 unary relations P_i to the language, with axioms stating that for every x exactly one of the P_i is true, the values of y for which P_i(y) is true are dense, and P₁ is true for all c_i. Then the models for which the sequence of elements c_i converge to a limit c split into n−2 cases depending on for which i the relation P_i(c) is true.

The idea of the proof of Vaught's theorem is as follows. If there are at most countably many countable models, then there is a smallest one: the atomic model, and a largest one, the saturated model, which are different if there is more than one model. If they are different, the saturated model must realize some n-type omitted by the atomic model. Then one can show that an atomic model of the theory of structures realizing this n-type (in a language expanded by finitely many constants) is a third model, not isomorphic to either the atomic or the saturated model. In the example above with 3 models, the atomic model is the one where the sequence is unbounded, the saturated model is the one where the sequence does not converge, and an example of a type not realized by the atomic model is an element greater than all elements of the sequence.