The singleton degrees of the Σ⁰₂ sets are not dense

2024/12/25 by Kent, Thomas F., Ng, Keng Meng, Sorbi, Andrea
#03D25 #03D30 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2412.18991

Abstract

Answering an open question raised by Cooper, we show that there exist Δ⁰₂ 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 Σ⁰₂ singleton degrees, and the Δ⁰₂ singleton degrees, are not dense (and consequently the Π⁰₂ Q-degrees, and the Δ⁰₂ Q-degrees, are not dense). Moreover D and E can be built to lie in the same enumeration degree.

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