A theory formulated in a countable predicate calculus can have at most 2א0 nonisomorphic countable models. In 1961 R. L. Vaught [9] conjected that if such a theory has uncountably many countable models, then it has exactly 2א0 countable models. This would of course follow immediately if one assumed the continuum hypothesis to be true. Almost ten years later, M. Morley [5] proved that if a count...

We construct Abelian group with an extra structure whose first order theory has finitely many but more than one countable model.

To clarify notation, Aut(A) is the group of all automorphisms of A and Th(Aut(A)) is the first-order theory of that group. An L-structure A = (A, . . .), where L is some first-order language, is resplendent (consult, for example, [Hod] or [Poi]) if, whenever L ⊇ L is a language that may have some constant symbols denoting elements of A and σ is an L-sentence that has a model B such that A 4 B↾L...

Abstract. A topological group H is called ω -narrow if for every neighbourhood V of it’s identity element there exists a countable set A such that V A = H = AV. A semigroup G is called a generalized group if for any x ∈ G there exists a unique element e(x) ∈ G such that xe(x) = e(x)x = x and for every x ∈ G there exists x − 1 ∈ G such that x − 1x = xx − 1 = e(x). Also le...

The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary prove theorems of ordinary mathematics, usually working in language second-order arithmetic L 2 . A major theme RM is therefore study structures that are countable or can be approximated by sets. Now, sets must represented sequences here, because higher-order definition ‘countable set’ involving injections/bij...

We characterize the ω-stable theories all of whose countable models admit decidable presentations. In particular, we show that for a countable ω-stable T , every countable model of T admits a decidable presentation if and only if all n-types in T are recursive and T has only countably many countable models. We further characterize the decidable models of ω-stable theories with countably many co...

We show that many countable groups acting on trees, including free products of infinitely and surface groups, are isomorphic to dense subgroups isometry bounded Urysohn spaces. This extends previous results the first t

1 Probability Theory over Countable Spaces Definition 1 (Countable Probability Space). A Countable Probability Space is composed of the tuple (Ω,Pr). • Countable Sample Space Ω The sample space is a set of outcomes for a particular experiment. “Countable” means that either the size of Ω is finite or there is a one-to-one and onto correspondence between the the elements of Ω and the set of Natur...

In Bue93b] we proved Vaught's conjecture for all superstable theories of nite rank; that is, such a theory has countably many or continuum many countable models. While this proof does settle Vaught's conjecture for unidimensional theories a sharper result can be obtained for these theories and in places the proof can be simpliied. Let T be a properly unidimensional theory with < 2 @ 0 many coun...