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^ℵ₀. Morley showed that number of countable models is finite or ℵ₀ or ℵ₁ or 2^ℵ₀, which solves the conjecture except for the case of ℵ₁ models when the continuum hypothesis fails. For this remaining case, Robin Knight (2002, 2007) has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. As of 2016 the counterexample has not been verified.

Let ${\displaystyle T}$ be a first-order, countable, complete theory with infinite models. Let ${\displaystyle I(T,\alpha )}$ denote the number of models of T of cardinality ${\displaystyle \alpha }$ up to isomorphism, the spectrum of the theory ${\displaystyle T}$. 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 ${\displaystyle \aleph _{0}<I(T,\aleph _{0})<2^{\aleph _{0}}}$. 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).

...
Wikipedia