Journal of Symbolic Logic

Lattice initial segments of the hyperdegrees

Bjørn Kjos-Hanssen and Richard A. Shore

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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.

Article information

Source
J. Symbolic Logic, Volume 75, Issue 1 (2010), 103-130.

Dates
First available in Project Euclid: 25 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1264433911

Digital Object Identifier
doi:10.2178/jsl/1264433911

Mathematical Reviews number (MathSciNet)
MR2605884

Zentralblatt MATH identifier
1194.03036

Citation

Shore, Richard A.; Kjos-Hanssen, Bjørn. Lattice initial segments of the hyperdegrees. J. Symbolic Logic 75 (2010), no. 1, 103--130. doi:10.2178/jsl/1264433911. https://projecteuclid.org/euclid.jsl/1264433911


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