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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2512.03778 |
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| _version_ | 1866908693350580224 |
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| author | Liu, Yong Peng, Cheng |
| author_facet | Liu, Yong Peng, Cheng |
| contents | 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 $\mathsf{I}Σ_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03778 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bi-Isolated d.c.e. Degrees and $Σ_1$ Induction Liu, Yong Peng, Cheng Logic 03D28, 03F30, 03H15 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 $\mathsf{I}Σ_1$. |
| title | Bi-Isolated d.c.e. Degrees and $Σ_1$ Induction |
| topic | Logic 03D28, 03F30, 03H15 |
| url | https://arxiv.org/abs/2512.03778 |