# The $\Sigma_2$ theory of $\mathscr{D}_h(\leq_h \mathcal{O})$ as an uppersemilattice with least and greatest element is decidable

## TL;DR

This paper proves that the $oldsymbol{
m ext{ extSigma}_2}$ theory of the hyperarithmetic degrees below Kleene's $oldsymbol{
m ext{ extO}}$ as an uppersemilattice with bounds is decidable, using novel initial-segment and embedding extension results.

## Contribution

It introduces new initial-segment and embedding extension theorems in the hyperarithmetic setting to establish decidability of the $oldsymbol{
m ext{ extSigma}_2}$ theory.

## Key findings

- Decidability of the $oldsymbol{
m ext{ extSigma}_2}$ theory established.
- New initial-segment result in hyperarithmetic degrees.
- New extension of embeddings result in hyperarithmetic degrees.

## Abstract

We establish the decidability of the $\Sigma_2$ theory of $\mathscr{D}_h(\leq_h \mathcal{O})$, the hyperarithmetic degrees below Kleene's $\mathcal{O}$, in the language of uppersemilattices with least and greatest element. This requires a new kind of initial-segment result and a new extension of embeddings result both in the hyperarithmetic setting.