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Lattice initial segments of the hyperdegrees

Published online by Cambridge University Press:  12 March 2014

Richard A. Shore  and
Bjørn Kjos-Hanssen
Richard A. Shore
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA, E-mail: [email protected]
Bjørn Kjos-Hanssen
Affiliation:
Department of Mathematics, University of Hawai'i at Mānoa, Honolulu, HI 96822, USA, E-mail: [email protected]
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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, . 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 ω1 ck . Somewhat surprisingly, the set theoretic analogof this forcing does not preserve ω1. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 1 , March 2010 , pp. 103 - 130
Copyright
Copyright © Association for Symbolic Logic 2010

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Footnotes

*

Partially supported by NSF Grants DMS-0554855 and DMS-0852811 and Grant 13408 from the John Templeton Foundation.

†

Partially supported by NSF grants DMS-0652669 and DMS-0901020. The authors also thank the referee for suggestions that improved the presentation in several ways.

References

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