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A Borel reducibility theory for classes of countable structures. (English) Zbl 0692.03022

Consider the class S(\(\rho)\) of all structures on \(\omega\) of similarity type \(\rho\) as an appropriate product of copies of the Cantor space. An invariant Borel class is the class of models of an \(L_{\omega_ 1\omega}\)-sentence. Let A and B be invariant subsets of S(\(\rho)\) and \(S(\rho ')\), respectively, A is Borel reducible to B, \(A\leq B\), iff there is a Borel measurable function \(\Phi\) : \(A\to B\) which is well-defined and one-to-one on isomorphism types. B is Borel complete iff B is Borel and \(A\leq B\) for all invariant Borel A. Besides some general results about these notions, the paper gives various mathematically natural classes with respect to \(\leq\). In particular, the class of linear orders and the class of fields of characteristic 0 are Borel complete.
Reviewer: J.Flum

MSC:

03C15 Model theory of denumerable and separable structures
03C75 Other infinitary logic
03D55 Hierarchies of computability and definability
03E15 Descriptive set theory
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