  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.

Logic