Morley’s theorem states that the number of non-isomorphic countable models of a complete countable first-order theory in a countable language is ℵ0 or ℵ1 or 2ℵ0. Vaught’s conjecture remains one of the most important open problems in Model Theory, asking whether ℵ1 can be omitted in the conclusion of Morley’s theorem. Even though Vaught’s conjecture is trivially false in second-order logic, no result was known regarding Morley’s trichotomy for second-order logic. We shall show using forcing, large cardinals and descriptive set theory that the second-order version of Morley’s theorem is undecidable.