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Vaught conjecture

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Vaught conjecture
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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{{sup|ℵ0}}. Morley showed that number of countable models is finite or ℵ0 or ℵ1 or 2{{sup|ℵ0}}, which solves the conjecture except for the case of ℵ1 models when the continuum hypothesis fails. For this remaining case, {{harvs|txt|first=Robin |last=Knight|year1=2002|year2=2007}} has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. As of 2016 the counterexample has not been verified.

Statement of the conjecture

Let T be a first-order, countable, complete theory with infinite models. Let I(T, alpha) denote the number of models of T of cardinality alpha up to isomorphism, the spectrum of the theory T. Morley proved that if I(T,ℵ0) 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 aleph_{0} < I(T,aleph_{0}) < 2^{aleph_{0}}. 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 which states that any countable complete T with uncountably many countable models will have a perfect set of uncountable models (as pointed out by John Steel, in On Vaught's conjecture. Cabal Seminar 76—77 (Proc. Caltech-UCLA Logic Sem., 1976—77), pp. 193–208, Lecture Notes in Math., 689, Springer, Berlin, 1978, this form of the Vaught conjecture is equiprovable with the original).

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 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


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