Distinguishing Models by Formulas and the Number of Countable Models

We indicate a way of distinguishing between structures, for which, we call two structures distinguishable. Roughly, being distinguishable means that they differ in the number of realizations each gives for some formula. Being non-distinguishable turns out to be an interesting equivalence relation that is weaker than isomorphism and stronger than elementary equivalence. We show that this equivalence relation is Borel in a Polish space that codes countable structures. It then follows, without assuming the Continuum Hypothesis, that for any first order theory in a countable language, if it has an uncountable set of countable models that are pairwise distinguishable, then actually it has such a set of continuum size. We show also, as an easy consequence of our results, that Vaught's conjecture holds for the language with only one unary relation symbol.