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.
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Support of DFG Grant Wu 101/10-1 and RFFI Grant 03-01-00757 acknowledged.
Received April 28, 2002; in final form January 24, 2003 Published online October 24, 2003
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Kanovei, V., Reeken, M. Borel and Countably Determined Reducibility in Nonstandard Domain. Monatsh. Math. 140, 197–231 (2003). https://doi.org/10.1007/s00605-003-0004-y
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DOI: https://doi.org/10.1007/s00605-003-0004-y