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Borel equivalence and isomorphism of coanalytic sets. (English) Zbl 0568.03023

The authors construct several interesting families of pairwise non-Borel- isomorphic coanalytic sets of reals assuming \(V=L\). Using the concept of recursive ordinals, admissible ordinals and decompositions are defined. Via admissible decompositions Borel equivalence is introduced. The examples are based on Theorem 5 claiming that under \(V=L\) Borel equivalence is necessary for Borel isomorphism. In Theorems 1 to 4 the existence of the non-Borel-isomorphic families of coanalytic subsets of \(N^ N\) is formulated under \(V=L\). The non-Borel-equivalence of the classes in Theorems 1 and 3 are not due to thin sets while Theorems 2 and 4 assert the existence of non-Borel-equivalent families of thin sets. Theorem 6 claims that if all projective games are determined then any two coanalytic non-Borel sets are Borel equivalent. Under the assumption that there exists a thin uncountable coanalytic set the existence of four non- Borel-isomorphic sets is proved.
Reviewer: P.Holicky

MSC:

03E15 Descriptive set theory
03E45 Inner models, including constructibility, ordinal definability, and core models
03D80 Applications of computability and recursion theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)