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The singleton degrees of the $\Sigma^0_2$ sets are not dense
- Publication Year :
- 2024
-
Abstract
- Answering an open question raised by Cooper, we show that there exist $\Delta^0_2$ 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 $\Sigma^{0}_{2}$ singleton degrees, and the $\Delta^{0}_{2}$ singleton degrees, are not dense (and consequently the $\Pi^0_2$ $Q$-degrees, and the $\Delta^{0}_{2}$ $Q$-degrees, are not dense). Moreover $D$ and $E$ can be built to lie in the same enumeration degree.
- Subjects :
- Mathematics - Logic
03D25, 03D30
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2412.18991
- Document Type :
- Working Paper