Borel Equivalence Relations

This article surveys the rapidly evolving area of Borel equivalence relations under the ordering of Borel reducibility. Although the field is now considered part of descriptive set theory, it traces its origins back to areas entirely outside logic. In fact this survey starts with Silver’s theorem on the number of equivalence classes of a co-analytic equivalence relation and the landmark Harrington-Kechris-Louveau dichotomy theorem, but also takes care to sketch some of the prehistory of the subject, going back to the roots in ergodic theory, dynamics, group theory, and functional analysis. In the later parts of the survey reader is also introduced to the theory of countable Borel equivalence relations and the connections with highly sophisticated techniques in superrigidity, the limits of structural theorems in the theory of Borel reducibility, turbulence and dynamical properties in the context of dichotomy theorems, and the connections with the theory of cardinality in the context of inner models of ZF which fail the axiom of choice.