The number of countable models via Algebraic logic

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 countable models, then T has 2^aleph_0 such models. Morley showed that if T has more than aleph_1 pairwise non isomorphic countable models, then it has 2^aleph_0 such models. In this paper, we First show how we can use algebraic logic, namely the representation theory of cylindric and quasi-polyadic algebras, to study Vaught's conjecture (count models), and we re-prove Morley's above mentioned theorem. Second, we show that Morley's theorem holds for the number of non isomorphic countable models omitting a countable family of types. We go further by giving examples showing that although this number can only take the values given by Morley's theorem, it can be different from the number of all non isomorphic countable models. Moreover, our examples show that the number of countable models omitting a family of types can also be either aleph_1 or 2 and therefore different from the possible values provided by Vaught's conjecture and by his well known theorem; in the case of aleph_1, however, the family is uncountable. Finally, we discuss an omitting types theorem of Shelah

Borel Complexity of the Isomorphism Relation for O-minimal Theories

In 1988, Mayer published a strong form of Vaught\u27s Conjecture for o-minimal theories (1). She showed Vaught\u27s Conjecture holds, and characterized the number of countable models of an o-minimal theory T if T has fewer than 2K ° countable models. Friedman and Stanley have shown in (2) that several elementary classes are Borel complete. This work addresses the class of countable models of an o-minimal theory T when T has 2N ° countable models, including conditions for when this class is Borel complete. The main result is as follows. Theorem 1. Let T be an o-minimal theory in a countable language having 2N ° countable models. Either i. For every finite set A, every p(x) E S1 (A) is simple, and isomorphism on the class of countable models of T is ∏03 (and is, in fact, equivalence of countable sets of reals); or ii. For some finite set A, some p(x) E S1 (A) is non-simple, and there is a finite set B D A such that the class of countable models of T over B is Borel complete