Vaught conjecture

The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ₀ or 2. Morley showed that the number of countable models is finite or ℵ₀ or ℵ₁ or 2, which solves the conjecture except for the case of ℵ₁ models when the continuum hypothesis fails. For this remaining case, has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. As of 2021, the counterexample has not been verified.

Statement of the conjecture

Let be a first-order, countable, complete theory with infinite models. Let denote the number of models of T of cardinality up to isomorphism—the spectrum of the theory. Morley proved that if I is infinite then it must be ℵ₀ or ℵ₁ or the cardinality of the continuum. The Vaught conjecture is the statement that it is not possible for. The conjecture is a trivial consequence of the continuum hypothesis; so this axiom is often excluded in work on the conjecture. Alternatively, there is a sharper form of the conjecture that states that any countable complete T with uncountably many countable models will have a perfect set of uncountable models.

Original formulation

The original formulation by Vaught was not stated as a conjecture, but as a problem: Can it be proved, without the use of the continuum hypothesis, that there exists a complete theory having exactly ℵ₁ non-isomorphic denumerable models? By the result by Morley mentioned at the beginning, a positive solution to the conjecture essentially corresponds to a negative answer to Vaught's problem as originally stated.

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 countably infinite 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, or the theory of a dense unbounded total order.
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 models 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 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 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 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 converges, and an example of a type not realized by the atomic model is an element greater than all elements of the sequence.

Topological Vaught conjecture

The topological Vaught conjecture is the statement that whenever a Polish group acts continuously on a Polish space, there are either countably many orbits or continuum many orbits. The topological Vaught conjecture is more general than the original Vaught conjecture: Given a countable language we can form the space of all structures on the natural numbers for that language. If we equip this with the topology generated by first-order formulas, then it is known from A. Gregorczyk, A. Mostowski, C. Ryll-Nardzewski, "Definability of sets of models of axiomatic theories" , vol. 9 that the resulting space is Polish. There is a continuous action of the infinite symmetric group that gives rise to the equivalence relation of isomorphism. Given a complete first-order theory T'', the set of structures satisfying T is a minimal, closed invariant set, and hence Polish in its own right.