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arXiv · 2512.03778

Bi-Isolated d.c.e. Degrees and $\Sigma_1$ Induction

Yong Liu·Cheng Peng
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Abstract

A Turing degree is d.c.e. if it contains a set that is the difference of two c.e. sets. A d.c.e. degree $\mathbf{d}$ is isolated if there exists a c.e. degree $\mathbf{a}<\mathbf{d}$ such that every c.e. degree below $\mathbf{d}$ is also below $\mathbf{a}$; $\mathbf{d}$ is upper isolated if there exists a c.e. degree $\mathbf{a}>\mathbf{d}$ such that every c.e. degree above $\mathbf{d}$ is also above $\mathbf{a}$; $\mathbf{d}$ is bi-isolated if it is both isolated and upper isolated. In this paper, we prove the existence of bi-isolated d.c.e. degrees in models of $\mathsf{I}\Sigma_1$.

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Yong Liu, Cheng Peng. 2025-12-03. Bi-Isolated d.c.e. Degrees and $\Sigma_1$ Induction. https://arxiv.org/abs/2512.03778

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