Non-Glimm–Effros equivalence relations at second projective level

Vladimir Kanovei

Fundamenta Mathematicae (1997)

  • Volume: 154, Issue: 1, page 1-35
  • ISSN: 0016-2736

Abstract

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A model is presented in which the Σ 2 1 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 2 1 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 o l d to ℵ 1 o l d .

How to cite

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Kanovei, Vladimir. "Non-Glimm–Effros equivalence relations at second projective level." Fundamenta Mathematicae 154.1 (1997): 1-35. <http://eudml.org/doc/212225>.

@article{Kanovei1997,
abstract = {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 $ℵ_1^\{old\}$.},
author = {Kanovei, Vladimir},
journal = {Fundamenta Mathematicae},
keywords = {equivalence relation; Glimm-Effros dichotomy; ill founded iterations of forcing; equiconstructibility relation; iteration of Sacks reals; generic extensions of },
language = {eng},
number = {1},
pages = {1-35},
title = {Non-Glimm–Effros equivalence relations at second projective level},
url = {http://eudml.org/doc/212225},
volume = {154},
year = {1997},
}

TY - JOUR
AU - Kanovei, Vladimir
TI - Non-Glimm–Effros equivalence relations at second projective level
JO - Fundamenta Mathematicae
PY - 1997
VL - 154
IS - 1
SP - 1
EP - 35
AB - 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 $ℵ_1^{old}$.
LA - eng
KW - equivalence relation; Glimm-Effros dichotomy; ill founded iterations of forcing; equiconstructibility relation; iteration of Sacks reals; generic extensions of
UR - http://eudml.org/doc/212225
ER -

References

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  12. [12] V. Kanovei, Ulm classification of analytic equivalence relations in generic universes, Math. Logic Quart. 44 (1998), to appear. Zbl0921.03048
  13. [13] A. S. Kechris, Topology and descriptive set theory, Topology Appl. 58 (1994), 195-222. Zbl0805.54035
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