Lattice initial segments of the hyperdegrees

We a¢ rm 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 …nite lattice) is isomorphic to an initial segment of Dh. Corollaries include the decidability of the two quanti…er theory of Dh and the undecidability of its three quanti…er 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 !CK 1 . Somewhat surprisingly, the set theoretic analog of this forcing does not preserve !1. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of Dh.