Borel and countably determined reducibility in nonstandard domain

Abstract

We consider reducibility of equivalence relations (ERs, for brevity), in a nonstandard domain, in terms of the Borel reducibility and the countably determined (CD, for brevity) reducibility. This reveals phenomena partially analogous to those discovered in descriptive set theory. The Borel reducibility structure of Borel sets and (partially) CD reducibility structure of CD sets in *N is described. We prove that all CD ERs with countable equivalence classes are CD-smooth, but not all are B-smooth, for instance, the ER of having finite difference on *N. Similarly to the Silver dichotomy theorem in Polish spaces, any CD ER on *N either has at most continuum-many classes or there is an infinite internal set of pairwise inequivalent elements. Our study of monadic ERs on *N, i.e., those of the form x E y iff |x-y| belongs to a given additive Borel cut in *N, shows that these ERs split in two linearly families, associated with countably cofinal and countably coinitial cuts, each of which is linearly ordered by Borel reducibility. The relationship between monadic ERs and the ER of finite symmetric difference on hyperfinite subsets of *N is studied.