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Non-distributive upper semilattice of Kleene degrees. (English) Zbl 0926.03048

Summary: \({\mathcal K}\) denotes the upper semilattice of all Kleene degrees. Under ZF+AD+DC, \({\mathcal K}\) is well-ordered and \(\deg(X^{SJ})\) is the next Kleene degree above \(\deg(X)\) for \(X\subseteq{^\omega \omega}\). While, without AD, properties of \({\mathcal K}\) are not always clear. In this note, we prove the non-distributivity of \({\mathcal K}\) under ZFC, and that of Kleene degrees between \(\deg(X)\) and \(\deg(X^{SJ})\) for some \(X\) under ZFC+CH.

MSC:

03D65 Higher-type and set recursion theory
03D30 Other degrees and reducibilities in computability and recursion theory
03E60 Determinacy principles
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References:

[1] Determinacy and type 2 recursion 36 pp 374– (1971)
[2] Habilitationsschrift (1984)
[3] Constructibility · Zbl 0542.03029
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