BOREL EQUIVALENCE-RELATIONS INDUCED BY ACTIONS OF THE SYMMETRICAL GROUP

We consider Borel equivalence relations E induced by actions of the in finite symmetric group, or equivalently the isomorphism relation on cl asses of countable models of bounded Scott rank. We relate the descrip tive complexity of the equivalence relation to the nature of its compl ete invariants. A typical theorem is that E is potentially Pi(3)(0) if f the invariants are countable sets of reals, it is potentially Pi(4)( 0) iff the invariants are countable sets of countable sets of reals, a nd so on. The proofs use various techniques, including Vaught transfor ms, changing topologies, and the Scott analysis of countable models. ( C) 1998 Published by Elsevier Science B.V. All rights reserved.