Abstract

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