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Glimm-Effros for coanalytic equivalence relations. (English) Zbl 1178.03065
The author proves the following result in the style of the Glimm-Effros dichotomy (the latter concerns Borel equivalence relations): If every real has a sharp, then any \(\Pi^1_1\) equivalence relation either Borel reduces \(E_0\) or in a \(\Delta^1_3\) manner allows the assignment of bounded subsets of \(\omega_1\) as complete invariants to the equivalence classes. (The analogous result for \(\Sigma^1_1\) equivalence relations was established earlier by the author and Kechris.) The author asks whether it is possible to replace \(\Delta^1_3\) by \(\Delta^1_2\), and whether the assumption of sharps is necessary.

03E15 Descriptive set theory
03E60 Determinacy principles
Full Text: DOI
[1] DOI: 10.1090/S0894-0347-1990-1057041-5
[2] DOI: 10.1016/0168-0072(94)00031-W · Zbl 0837.03040
[3] Admissible sets and structures (1975)
[4] Descriptive set theory 100 (1980) · Zbl 0433.03025
[5] Model theory for infinitary logic. Logic with countable conjunctions and finite quantifiers 62 (1971) · Zbl 0222.02064
[6] Logic Colloquium ’80 (Prague, 1980) 108 pp 147– (1982)
[7] DOI: 10.1016/S0168-0072(99)00013-5 · Zbl 0942.03055
[8] Memoirs of the American Mathematical Society 140 (1999)
[9] DOI: 10.2307/421148 · Zbl 0889.03038
[10] Analytic equivalence relations and Vim-type classifications 60 pp 1273– (1995)
[11] Set theory (1978)
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