Lattice initial segments of the hyperdegrees

Abstract

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, 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 𝒟_h. Corollaries include the decidability of the two quantifier theory of 𝒟_h 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 𝒟_h.