# Non-Glimm–Effros equivalence relations at second projective level

Fundamenta Mathematicae (1997)

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

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topKanovei, 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 -

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