Borel equivalence relations and classifications of countable models

Greg Hjorth, Alexander S. Kechris
1996 Annals of Pure and Applied Logic  
Using the theory of Bore1 equivalence relations we analyze the isomorphism relation on the countable models of a theory and develop a framework for measuring the complexity of possible complete invariants for isomorphism. In~~uction (A) Let L be a first order language and X some class of countable L-structures. In this paper we will be exclusively concerned with the case that X consists of the countable models of some (T EL", -such as the collection of countable abelian torsion-free groups,
more » ... table fields, finitely generated groups, connected locally finite graphs, and so on. In this abstract setting we consider what types of complete invariants can be used to classify the elements of x up to isomorphism. We use the methods and concepts of the general theory of Bore1 equivalence relations to provide an analysis of the isomorphism relation and a framework for measuring the complexity of possible invariants. We denote by XL the space of co~~ble s~ctures of L with universe N = (0, 1, 2,. . .}. These represent, up to isomorphism, all infinite countable models of L. Since we are only interested here in infinite models, "countable model" will mean "infinite countable model" in this paper. The space XL is a Polish space under a natural topology. Denote by Z the isomorphism relation on XL and for every theory (r E L" w let Mod(n) = {JZ EX~, : x + CT}. Then Mod(o) is an isomorphism-invariant Bore1 subset of XL and, by a theorem of Lopez and Escobar, any such set is of the form Mod(a) for some 0 E L" cu. We also denote the restriction of GZ to Mod(o) by %.
doi:10.1016/s0168-0072(96)00006-1 fatcat:ukbcsqjojjfyphfefc77yhovra