Analytic Equivalence Relations Satisfying Hyperarithmetic-is-recursive
نویسنده
چکیده
We prove, in ZF+Σ2-determinacy, that for any analytic equivalence relation E, the following three statements are equivalent: (1) E does not have perfectly many classes, (2) E satisfies hyperarithmetic-is-recursive on a cone, and (3) relative to some oracle, for every equivalence class [Y ]E we have that a real X computes a member of the equivalence class if and only if ω 1 ≥ ω [Y ] 1 . We also show that the implication from (1) to (2) is equivalent to the existence of sharps over ZF .
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تاریخ انتشار 2013