Vaught's Theorem

Vaught's Theorem states that a complete first-order theory with only infinite models can never have exactly two countable models up to isomorphism, though it may have exactly one, or three, or more, or infinitely many. Named for Robert Lawson Vaught, it is a classical result of model theory illustrating that the number of countable models a theory admits is constrained in subtle, non-obvious ways, a question the related Vaught conjecture pursues further for uncountably many models.