Journal of Symbolic Logic

Non-Distributive Upper Semilattice of Kleene Degrees

Hisato Muraki

Abstract

$\mathscr{K}$ 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).

Article information

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745697

Mathematical Reviews number (MathSciNet)
MR1683900

Zentralblatt MATH identifier
0926.03048

JSTOR