Forcing with Δ perfect trees and minimal Δ-degrees

Abstract

This paper is a sequel to [3] and it contains, among other things, proofs of the results announced in the last section of that paper. In §1, we use the general method of [3] together with reflection arguments to study the properties of forcing with Δ perfect trees, for certain Spector pointclasses Γ, obtaining as a main result the existence of a continuum of minimal Δ-degrees for such Γ's, under determinacy hypotheses. In particular, using PD, we prove the existence of continuum many minimal Δ^(1)_(2n+1)-degrees, for all n.^(2) Following an idea of Leo Harrington, we extend these results in §2 to show the existence of minimal strict upper bounds for sequences of Δ-degrees which are not too far apart. As a corollary, it is computed that the length of the natural hierarchy of Δ^(1)_(2n+1)-degrees is equal to ω when n ≥ 1. (By results of Sacks and Richter the length of the natural hierarchy of Δ^(1)_(1)-degrees is known to be equal to the first recursively inaccessible ordinal.)

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