Non-Glimm–Effros equivalence relations at second projective level
Volume 154 / 1997
Fundamenta Mathematicae 154 (1997), 1-35
DOI: 10.4064/fm-154-1-1-35
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 $ℵ_1^{old}$.
