Borel and Countably Determined Reducibility in Nonstandard Domain

Abstract.

We consider, in a nonstandard domain, reducibility of equivalence relations in terms of the Borel reducibility ≤_B and the countably determined (CD, for brevity) reducibility ≤_CD. This reveals phenomena partially analogous to those discovered in modern “standard” descriptive set theory. The ≤_CD-structure of CD sets (partially) and the ≤_B-structure of Borel sets (completely) in *ℕ are described. We prove that all “countable” (i.e., those with countable equivalence classes) CD equivalence relations (ERs) are CD-smooth, but not all are B-smooth: the relation x  M _ℕ y iff ∣x−y∣∈ℕ is a counterexample. Similarly to the Silver dichotomy theorem in Polish spaces, any CD equivalence relation on *ℕ either has at most continuum-many classes (and this can be witnessed, in some manner, by a countably determined function) or there is an infinite internal set of pairwise inequivalent elements. Our study of monadic equivalence relations, i.e., those of the form x  M _U   y iff ∣x−y∣∈U, where U is an additive countably determined cut (initial segment) demonstrates that these ERs split in two linearly ≤_B-(pre)ordered families, associated with countably cofinal and countably coinitial cuts. The equivalence u  FD  v iff uΔv is finite, on the set of all hyperfinite subsets of *ℕ, ≤_B-reduces all “countably cofinal” ERs but does not ≤_CD-reduce any of “countably coinitial” ERs.