On the decidability of the $\Sigma_2$ theories of the arithmetic and hyperarithmetic degrees as uppersemilattices

TL;DR

This paper proves that the $ ext{Sigma}_2$ theories of the arithmetic and hyperarithmetic degrees, viewed as uppersemilattices, are decidable by showing embeddings extend precisely when structures are end-extensions.

Contribution

It establishes the decidability of the $ ext{Sigma}_2$ theories for these degree structures using Kumabe-Slaman forcing and extension characterizations.

Findings

Decidability of $ ext{Sigma}_2$ theories for arithmetic and hyperarithmetic degrees.

Embeddings extend iff the structure is an end-extension.

Utilizes Kumabe-Slaman forcing to achieve results.

Abstract

We establish the decidability of the $\Sigma_2$ theory of both the arithmetic and hyperarithmetic degrees in the language of uppersemilattices i.e. the language with $\leq, 0$ and $\sqcup$. This is achieved by using Kumabe-Slaman forcing - along with other known results - to show that given finite uppersemilattices $\mathcal{M}$ and $\mathcal{N}$, where $\mathcal{M}$ is a subuppersemilattice of $\mathcal{N}$, then for both degree structures, every embedding of $\mathcal{M}$ into the structure extends to one of $\mathcal{N}$ iff $\mathcal{N}$ is an end-extension of $\mathcal{M}$.