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1997 | 154 | 1 | 1-35
Tytuł artykułu

Non-Glimm–Effros equivalence relations at second projective level

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A model is presented in which the $Σ^1_2$ equivalence relation xCy iff L[x]=L[y] of equiconstructibility of reals does not admit a reasonable form of the Glimm-Effros theorem. The model is a kind of iterated Sacks generic extension of the constructible model, but with an "ill"founded "length" of the iteration. In another model of this type, we get an example of a ${Π}^1_2$ non-Glimm-Effros equivalence relation on reals. As a more elementary application of the technique of "ill"founded Sacks iterations, we obtain a model in which every nonconstructible real codes a collapse of a given cardinal $κ ≥ ℵ_2^{old}$ to $ℵ_1^{old}$.
Słowa kluczowe
Rocznik
Tom
154
Numer
1
Strony
1-35
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-01-15
poprawiono
1997-01-20
Twórcy
  • Department of Mathematics, Moscow Transport Engineering Institute (MIIT), Obraztsova 15, Moscow 101475, Russia, kanovei@mech.math.msu.su
Bibliografia
  • [1] J. E. Baumgartner and R. Laver, Iterated perfect set forcing, Ann. Math. Logic 17 (1979), 271-288.
  • [2] S. D. Friedman and B. Velickovic, Nonstandard models and analytic equivalence relations, Proc. Amer. Math. Soc., to appear.
  • [3] M. Groszek, $ω_1*$ as an initial segment of the c-degrees, J. Symbolic Logic 59 (1994), 956-976.
  • [4] M. Groszek and T. Jech, Generalized iteration of forcing, Trans. Amer. Math. Soc. 324 (1991), 1-26.
  • [5] L. A. Harrington, A. S. Kechris and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), 903-928.
  • [6] G. Hjorth, Thin equivalence relations and effective decompositions, J. Symbolic Logic 58 (1993), 1153-1164.
  • [7] G. Hjorth, A dichotomy for the definable universe, J. Symbolic Logic 60 (1995), 1199-1207.
  • [8] G. Hjorth, A remark on $∏1^1$ equivalence relations, note, 1994.
  • [9] G. Hjorth and A. S. Kechris, Analytic equivalence relations and Ulm-type classifications, J. Symbolic Logic 60 (1995), 1273-1300.
  • [10] V. Kanovei, The cardinality of the set of Vitali equivalence classes, Math. Notes 49 (1991), 370-374.
  • [11] V. Kanovei, An Ulm-type classification theorem for equivalence relations in Solovay model, J. Symbolic Logic 62 (1997), to appear.
  • [12] V. Kanovei, Ulm classification of analytic equivalence relations in generic universes, Math. Logic Quart. 44 (1998), to appear.
  • [13] A. S. Kechris, Topology and descriptive set theory, Topology Appl. 58 (1994), 195-222.
  • [14] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995.
  • [15] N. Lusin, Sur les ensembles analytiques, Fund. Math. 10 (1927), 1-95.
  • [16] W. Sierpiński, L'axiome de M. Zermelo et son rôle dans la théorie des ensembles et l'analyse, Bull. Internat. Acad. Sci. Lettres Sér. A Sci. Math. 1918, 97-152.
  • [17] J. Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann. Math. Logic 18 (1980), 1-28.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv154i1p1bwm
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