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The singleton degrees of the $\Sigma^0_2$ sets are not dense

Authors :
Kent, Thomas F.
Ng, Keng Meng
Sorbi, Andrea
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

Subjects :
Mathematics - Logic
03D25, 03D30

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2412.18991
Document Type :
Working Paper