We affirm 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 finite
lattice) is isomorphic to an initial segment of Dh. 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 ω1CK. 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 an
initial segment of Dh