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 2ℵ0. Morley showed that number of countable models is finite or ℵ0 or ℵ1 or 2ℵ0, which solves the conjecture except for the case of ℵ1 models when the continuum hypothesis fails. For this remaining case, R. Knight (2002, 2007) has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture.
Read more about Vaught Conjecture: Statement of The Conjecture, Vaught's Theorem, Topological Vaught Conjecture
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