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Lattice initial segments of the hyperdegrees

  • Richard A. Shore (a1) and Bjørn Kjos-Hanssen (a2)

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 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 .

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Lattice initial segments of the hyperdegrees

  • Richard A. Shore (a1) and Bjørn Kjos-Hanssen (a2)

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