Borel equivalence relations induced by actions of the symmetric group

Annals of Pure and Applied Logic 92 (1):63-112 (1998)

Abstract

We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially Π03 iff the invariants are countable sets of reals, it is potentially Π04 iff the invariants are countable sets of countable sets of reals, and so on. The proofs use various techniques, including Vaught transforms, changing topologies, and the Scott analysis of countable models

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References found in this work

New Dichotomies for Borel Equivalence Relations.Greg Hjorth & Alexander S. Kechris - 1997 - Bulletin of Symbolic Logic 3 (3):329-346.
An Absoluteness Principle for Borel Sets.Greg Hjorth - 1998 - Journal of Symbolic Logic 63 (2):663-693.
Higher Set Theory and Mathematical Practice.Harvey M. Friedman - 1971 - Annals of Mathematical Logic 2 (3):325.

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