Morley's theorem and Vaught's conjecture via Algebraic logic

Abstract: Vaught's Conjecture states that if $T$ is a complete first order theory in a countable language that has more than $\aleph_0$ pairwise non-isomorphic countably infinite models, then $T$ has $2^{\aleph_0}$ such models. Morley showed that if $T$ has more than $\aleph_1$ pairwise non-isomorphic countably infinite models, then it has $2^{\aleph_0}$ such models.\ In this paper, we re-prove Morley's result and prove the corresponding statement for languages without equality, and for theories which are not necessarily complete. Our proof uses algebraic logic, namely the representation theory of cylindric and quasi-polyadic algebras. Also, as in Morley's proof, we use results from descriptive set theory. After all this, we show that our proof can be modified to talk about the number of models omitting a certain family of types. \end{abstract}