Lattice initial segments of the hyperdegrees

TL;DR

This paper proves that every countable distributive lattice can be embedded as an initial segment of the hyperdegrees, establishing a deep connection between lattice theory and hyperarithmetic degrees with implications for decidability.

Contribution

It introduces a new lattice representation theorem and forcing method to embed countable lattices into the hyperdegrees, confirming a conjecture of Sacks.

Findings

Every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees.

The two quantifier theory of the hyperdegrees is decidable.

The three quantifier theory of the hyperdegrees is undecidable.

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 $% \mathcal{D}_{h}$ 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 $\omega _{1}^{CK}$. 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 $\mathcal{D}_{h}$.