299 research outputs found
Beyond Borel-amenability: scales and superamenable reducibilities
We analyze the degree-structure induced by large reducibilities under the
Axiom of Determinacy. This generalizes the analysis of Borel reducibilities
given in references [1], [6] and [5] e.g. to the projective levels.Comment: 13 page
Inadmissible forcing
AbstractA structure is E-closed if it is closed under all partial E-recursive functions from V into V, a set theoretic extension of Kleene's partial recursive functions of finite type in the normal case. Let L(κ) be E-closed and ∑1 inadmissible. Then L(κ) has reflection properties useful in the study of generic extensions of L(κ). Every set generic extension of L(κ) via countably closed forcing conditions is E-closed. A class generic construction shows: if L(κ) is countable, and inside L(κ) the greatest cardinal gc(κ), has uncountable cofinality, then there exists a T ⊆ gc(κ) such that L(κ, T) = E(T), the least E-closed set with T as a member. A partial converse is obtained via a selection theorem that implies E(X) is ∑1 admissible when X is a set of ordinals and the greatest cardinal in the sense of E(X) has countable cofinality in E(X)
On the Relative Consistency Strength of Determinacy Hypothesis
For any collection of sets of reals C, let C-DET be the statement that all sets of reals in C are determined. In this paper we study questions of the form: For given C ⊆ C', when is C'-DET equivalent, equiconsistent or strictly stronger in consistency strength than C-DET (modulo ZFC)? We focus especially on classes C contained in the projective sets
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