On a Glimm – Effros dichotomy and an Ulm–type classification in Solovay model
نویسنده
چکیده
We prove that in Solovay model every OD equivalence E on reals either admits an OD reduction to the equality on the set of all countable (of length < ω1 ) binary sequences, or continuously embeds E0, the Vitali equivalence. If E is a Σ1 1 (resp. Σ 1 2 ) relation then the reduction in the “either” part can be chosen in the class of all ∆1 (resp. ∆2 ) functions. The proofs are based on a topology generated by OD sets.
منابع مشابه
Non-Glimm–Effros equivalence relations at second projective level
A model is presented in which the Σ1 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 Π1 2 non-Glimm–Effros equivalenc...
متن کاملMeasure Reducibility of Countable Borel Equivalence Relations
We show that every basis for the countable Borel equivalence relations strictly above E0 under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base, using arguments substantially simpler than those previously employed.
متن کاملOn a Glimm – Effros dichotomy theorem for Souslin relations in generic universes
We prove that if every real belongs to a set generic extension of the constructible universe then every Σ1 equivalence E on reals either admits a ∆1 reduction to the equality on the set 21 of all countable binary sequences, or continuously embeds E0, the Vitali equivalence. The proofs are based on a topology generated by OD sets.
متن کاملVaught’s Conjecture and the Glimm-effros Property for Polish Transformation Groups
We extend the original Glimm-Effros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught
متن کاملDefinable cardinals just beyond R/Q
We establish the inexistence of countable bases for the family of definable cardinals associated with countable Borel equivalence relations which are not measure reducible to E0, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push the primary known results concerning the abstract structure of the Borel cardinal hierarchy nearly to its base, using arguments sub...
متن کامل