The singleton degrees of the $Σ^0_2$ sets are not dense
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929648571514880 |
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| author | Kent, Thomas F. Ng, Keng Meng Sorbi, Andrea |
| author_facet | Kent, Thomas F. Ng, Keng Meng Sorbi, Andrea |
| contents | Answering an open question raised by Cooper, we show that there exist $Δ^0_2$ sets $D$ and $E$ such that the singleton degree of $E$ is a minimal cover of the singleton degree of $D$. This shows that the $Σ^{0}_{2}$ singleton degrees, and the $Δ^{0}_{2}$ singleton degrees, are not dense (and consequently the $Π^0_2$ $Q$-degrees, and the $Δ^{0}_{2}$ $Q$-degrees, are not dense). Moreover $D$ and $E$ can be built to lie in the same enumeration degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The singleton degrees of the $Σ^0_2$ sets are not dense Kent, Thomas F. Ng, Keng Meng Sorbi, Andrea Logic 03D25, 03D30 Answering an open question raised by Cooper, we show that there exist $Δ^0_2$ sets $D$ and $E$ such that the singleton degree of $E$ is a minimal cover of the singleton degree of $D$. This shows that the $Σ^{0}_{2}$ singleton degrees, and the $Δ^{0}_{2}$ singleton degrees, are not dense (and consequently the $Π^0_2$ $Q$-degrees, and the $Δ^{0}_{2}$ $Q$-degrees, are not dense). Moreover $D$ and $E$ can be built to lie in the same enumeration degree. |
| title | The singleton degrees of the $Σ^0_2$ sets are not dense |
| topic | Logic 03D25, 03D30 |
| url | https://arxiv.org/abs/2412.18991 |