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NOTES ON SACKS’ SPLITTING THEOREM
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 89 / Issue 4 / December 2024
- Published online by Cambridge University Press:
- 26 October 2023, pp. 1768-1797
- Print publication:
- December 2024
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- Article
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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 
$A_2$.
