NOTES ON SACKS’ SPLITTING THEOREM Article Swipe

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Klaus Ambos‐Spies , Rod Downey , Martin Monath , KENG MENG NG ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.1017/jsl.2023.77 · OA: W4387958294

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 $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$ -c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $ -c.a. as shown in [9]. We also show that if cone avoidance is added then there is no level below $\varepsilon _0$ which can be used to characterize the complexity of $A_1$ and $A_2$ .