Journal of Symbolic Logic

Non-Distributive Upper Semilattice of Kleene Degrees

Hisato Muraki

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$\mathscr{K}$ denotes the upper semilattice of all Kleene degrees. Under ZF + AD + DC, $\mathscr{K}$ is well-ordered and deg(X$^{SJ}$) is the next Kleene degree above deg(X) for $X \subseteq\omega\omega$ (see [4] and [5. Chapter V]). While, without AD, properties of $\mathscr{K}$ are not always clear. In this note, we prove the non-distributivity of $\mathscr{K}$ under ZFC ($\S$1), and that of Kleene degrees between deg(X) and deg(X$^{SJ}$) for some X under ZFC + CH ($\S$2,3).

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J. Symbolic Logic, Volume 64, Issue 1 (1999), 147-158.

First available in Project Euclid: 6 July 2007

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Muraki, Hisato. Non-Distributive Upper Semilattice of Kleene Degrees. J. Symbolic Logic 64 (1999), no. 1, 147--158.

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