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Classification from a computable viewpoint
 The Bulletin of Symbolic Logic
"... Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism ..."
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Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism
Vaught's Conjecture for Unidimensional Theories
, 1994
"... In [Bue93b] we proved Vaught's conjecture for all superstable theories of finite 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 th ..."
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In [Bue93b] we proved Vaught's conjecture for all superstable theories of finite 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 simplified. Let T be a properly unidimensional theory with ! 2 @ 0 many countable models. First, we prove for T what is called the Tree Theorem.
Finitely axiomatizable strongly minimal groups
, 2007
"... We show that if G is a strongly minimal finitely axiomatizable group, the division ring of quasiendomorphisms of G must be an infinite finitely presented ring. Questions about finite axiomatizability of first order theories are nearly as old as model theory itself and seem at first glance to have a ..."
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We show that if G is a strongly minimal finitely axiomatizable group, the division ring of quasiendomorphisms of G must be an infinite finitely presented ring. Questions about finite axiomatizability of first order theories are nearly as old as model theory itself and seem at first glance to have a fairly syntactical flavor. But it was in order to show that totally categorical theories cannot be finitely axiomatized that, in the early eighties, Boris Zilber started developing what is now known as “Geometric stability theory”. Indeed, as is often the case, in order to answer such a question, one needs to develop a fine analysis of the structure of models in the class involved and to understand exactly how each model is constructed. The easiest way to force a structure to be infinite by one first order sentence is to impose an ordering without end points, or a dense ordering, thus making the structure unstable. It was hence rather natural to wonder about theories at the other extremity of the stability spectrum, and in the early 60’s the question was posed whether there existed
The Classification of Small Types of Rank, part I
, 1998
"... Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we prove Theorem A. Le ..."
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Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we prove Theorem A. Let G be a superstable group of U rank # such that the generics of G are locally modular and Th(G) has few countable models. Let G  be the group of nongeneric elements of G , G + = G o + G  . Let # = { q # S(#) : U(q # #) < # } . For any countable model M of Th(G) there is a finite A # M such that M is almost atomic over A#G + (M)# # p## p(M) . 1