Friedman, Harvey; Stanley, Lee A Borel reducibility theory for classes of countable structures. (English) Zbl 0692.03022 J. Symb. Log. 54, No. 3, 894-914 (1989). 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 Cited in 6 ReviewsCited in 126 Documents MSC: 03C15 Model theory of denumerable and separable structures 03C75 Other infinitary logic 03D55 Hierarchies of computability and definability 03E15 Descriptive set theory Keywords:Borel reducibility; Borel completeness; Cantor space; invariant Borel class; Borel-measurable function; linear orders; fields of characteristic 0 PDFBibTeX XMLCite \textit{H. Friedman} and \textit{L. Stanley}, J. Symb. Log. 54, No. 3, 894--914 (1989; Zbl 0692.03022) Full Text: DOI References: [1] Infinite Abelian groups 2 (1970) [2] DOI: 10.1016/0001-8708(81)90021-9 · Zbl 0483.03030 · doi:10.1016/0001-8708(81)90021-9 [3] DOI: 10.1007/BFb0080977 · doi:10.1007/BFb0080977 [4] Hodge cycles, motives, and Shimura varieties 900 (1982) · Zbl 0465.00010 [5] DOI: 10.1007/BFb0069301 · doi:10.1007/BFb0069301 [6] Harvey Friedman’s research on the foundations of mathematics pp 11– (1985) [7] DOI: 10.1090/S0002-9947-1968-0244049-7 · doi:10.1090/S0002-9947-1968-0244049-7 [8] Stability of nilpotent groups of class 2 and prime exponent 46 pp 781– (1981) · Zbl 0482.03014 [9] Diophantine geometry (1962) [10] On definable subsets of p-adic fields 41 pp 605– (1976) [11] Introduction to algebraic geometry (1958) · Zbl 0095.15301 [12] Algebra (1967) [13] Infinite Abelian groups (1954) · Zbl 0057.01901 [14] DOI: 10.1007/BF02757141 · Zbl 0304.02025 · doi:10.1007/BF02757141 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.