Notes on Sacks' splitting theorem

• Verfasst von: Ambos-Spies, Klaus [VerfasserIn]
• Verfasst von: Downey, Rod G. [VerfasserIn]
• Verfasst von: Monath, Martin [VerfasserIn]
• Verfasst von: Ng, Keng Meng [VerfasserIn]
• Titel: Notes on Sacks' splitting theorem
• Verf.angabe: Klaus Ambos-Spies, Rod G. Downey, Martin Monath, and Keng Meng Ng
• E-Jahr: 2023
• Jahr: 26 October 2023
• Umfang: 30 S.
• Fussnoten: Gesehen am 26.03.2024
• Titel Quelle: Enthalten in: The journal of symbolic logic
• Ort Quelle: Cambridge : Cambridge Univ. Press, 1936
• Jahr Quelle: 2023
• Band/Heft Quelle: (2023), online ahead of print
• ISSN Quelle: 1943-5886
• DOI: doi:10.1017/jsl.2023.77
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 03D25
• Sach-SW: 03D55
• Sach-SW: Downey-Greenberg Hierarchy
• Sach-SW: Sacks Splitting Theorem
• Sach-SW: totally ω-c.a
• Sach-SW: unbounded finite injury
• K10plus-PPN: 1884364535

Abstract

We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets A0⊔A1=AA0⊔A1=AA_0\sqcup A_1=A splitting A with A0A0A_0 and A1A1A_1 both totally ω2ω2\omega ^2-c.a. in terms of the Downey-Greenberg hierarchy, and this result cannot be improved to totally ωω\omega -c.a. as shown in [9]. We also show that if cone avoidance is added then there is no level below ε0ε0\varepsilon _0 which can be used to characterize the complexity of A1A1A_1 and A2A2A_2.