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, ${\mathcal{D}}_h$. 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 ${\mathcal{D}}_h$. 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 ${\omega }_1^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ${\omega }_1$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of ${\mathcal{D}}_h$.

References in corpus (2)

• Lattice initial segments of the Turing degrees
• Local initial segments of the Turing degrees