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March 2010 Lattice initial segments of the hyperdegrees
Bjørn Kjos-Hanssen, Richard A. Shore
J. Symbolic Logic 75(1): 103-130 (March 2010). DOI: 10.2178/jsl/1264433911

Abstract

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, 𝒟h. 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 𝒟h. Corollaries include the decidability of the two quantifier theory of 𝒟h 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 ω₁CK. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω₁. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of 𝒟h.

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Bjørn Kjos-Hanssen. Richard A. Shore. "Lattice initial segments of the hyperdegrees." J. Symbolic Logic 75 (1) 103 - 130, March 2010. https://doi.org/10.2178/jsl/1264433911

Information

Published: March 2010
First available in Project Euclid: 25 January 2010

zbMATH: 1194.03036
MathSciNet: MR2605884
Digital Object Identifier: 10.2178/jsl/1264433911

Rights: Copyright © 2010 Association for Symbolic Logic

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Vol.75 • No. 1 • March 2010
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