The completeness of isomorphism

Abstract

In classical descriptive set theory, analytic equivalence relations (i.e., Σ1 equivalence relations with parameters) are compared under the relation of Borel reducibility (for example, see [5]). An important subclass of the Σ1 equivalence relations are the isomorphism relations, i.e., the restrictions of the isomorphism relation on countable structures (viewed as an equivalence relation on reals coding such structures) to the models of a sentence of the infinitary logic Lω1ω. Scott’s Theorem implies that the equivalence classes of any isomorphism relation are Borel, and therefore no isomorphism relation can be complete (under Borel reducibility) within the class of Σ1 equivalence relations as a whole, some of which contain non-Borel equivalence classes. (This is clarified below.)