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(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 . 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).
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