Vaught’s Conjecture Without Equality

Abstract

Suppose that $\sigma \in {\mathcal{L}}_{{\omega }_1,\omega }\left(\mathrm{L}\right)$ is such that all equations occurring in $\sigma$ are positive, have the same set of variables on each side of the equality symbol, and have at least one function symbol on each side of the equality symbol. We show that $\sigma$ satisfies Vaught’s conjecture. In particular, this proves Vaught’s conjecture for sentences of ${\mathcal{L}}_{{\omega }_1,\omega }\left(\mathrm{L}\right)$ without equality.