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arXiv · 1606.07392

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

James Barnes
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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 $\mathcal{M}$.

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James Barnes. 2016-06-23. On the decidability of the $\Sigma_2$ theories of the arithmetic and hyperarithmetic degrees as uppersemilattices. https://arxiv.org/abs/1606.07392

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