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  1. Lattice initial segments of the hyperdegrees.Richard A. Shore & Bjørn Kjos-Hanssen - 2010 - Journal of Symbolic Logic 75 (1):103-130.
    We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\scr{D}_{h}$ . 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 $\scr{D}_{h}$ . Corollaries include the decidability of the two quantifier theory of $\scr{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is a (...)
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