Definable cardinals just beyond R/Q
نویسندگان
چکیده
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 substantially simpler than those previously employed. Our main tool is a strong notion of separability, which holds of orbit equivalence relations induced by group actions satisfying an appropriate measureless local rigidity property.
منابع مشابه
A Generalization of the Gap Forcing Theorem
The Main Theorem of this article asserts in part that if an extension V ⊆ V satisfies the δ approximation and covering properties, then every embedding j : V → N definable in V with critical point above δ is the lift of an embedding j ↾ V : V → N definable in the ground model V . It follows that in such extensions there can be no new weakly compact cardinals, totally indescribable cardinals, st...
متن کاملOn equivalence relations second order definable over H(κ)
Let κ be an uncountable regular cardinal. Call an equivalence relation on functions from κ into 2 second order definable over H(κ) if there exists a second order sentence φ and a parameter P ⊆ H(κ) such that functions f and g from κ into 2 are equivalent iff the structure 〈H(κ),∈, P, f, g〉 satisfies φ. The possible numbers of equivalence classes of second order definable equivalence relations c...
متن کاملLarge Cardinals and Definable Well-Orderings of the Universe
We use a reverse Easton forcing iteration to obtain a universe with a definable wellordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ♦∗ κ+ at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardin...
متن کاملLarge cardinals and definable well-orders on the universe
We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ♦∗ κ+ at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal...
متن کاملDefinable Tree Property Can Hold at All Uncountable Regular Cardinals
Starting from a supercompact cardinal and a measurable above it, we construct a model of ZFC in which the definable tree property holds at all uncountable regular cardinals. This answers a question from [1]
متن کامل