28 research outputs found
Lattice initial segments of the hyperdegrees
We affirm a conjecture of Sacks [1972] by showing that every countable
distributive lattice is isomorphic to an initial segment of the hyperdegrees,
. In fact, we prove that every sublattice of any
hyperarithmetic lattice (and so, in particular, every countable locally finite
lattice) is isomorphic to an initial segment of . Corollaries
include the decidability of the two quantifier theory of
and the undecidability of its three quantifier theory. The key tool in the
proof is a new lattice representation theorem that provides a notion of forcing
for which we can prove a version of the fusion lemma in the hyperarithmetic
setting and so the preservation of . Somewhat surprisingly,
the set theoretic analog of this forcing does not preserve . On
the other hand, we construct countable lattices that are not isomorphic to an
initial segment of
The Theory of Countable Analytical Sets
The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable ∏_n^1 set of reals, C_n (this is also true for n even, replacing ∏_n^1 by Σ_n^1 and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets C_n is then investigated in detail, the point of departure being the fact that each C_n is a set of Δ_n^1-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, ω-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc
Martin's conjecture for regressive functions on the hyperarithmetic degrees
We answer a question of Slaman and Steel by showing that a version of
Martin's conjecture holds for all regressive functions on the hyperarithmetic
degrees. A key step in our proof, which may have applications to other cases of
Martin's conjecture, consists of showing that we can always reduce to the case
of a continuous function.Comment: 12 page
Forcing with Δ perfect trees and minimal Δ-degrees
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.