In Zermelo-Fraenkel set theory without the Axiom of Foundation we study the schema version of the principle of dependent choices in connection with Aczel's antifoundation axiom , Boffa's anti-foundation axiom, and axiom of collection.

Concerning Post's problem for Kleene degrees and its relativization, Hrbacek showed in [1] and [2] that if V = L, then Kleene degrees of coanalytic sets are dense, and then for all K ⊆ωω, there are N1 sets which are Kleene semirecursive in K and not Kleene recursive in each other and K. But the density of Kleene semirecursive in K Kleene degrees is not obtained from these theorems. In this note, we extend these theorems by showing that if V (...) = L, then for all K ⊆ ωω, Kleene semirecursive in K Kleene degrees are dense above K. (shrink)

In Aczel [1], the existence of largest fixed points of set continuous operators is proved assuming the schema version of dependent choices in Zermelo-Fraenkel set theory without the axiom of Foundation. In the present paper, we study whether the existence of largest fixed points of set continuous operators is provable without the schema version of dependent choices, using Boffa's weak antifoundation axioms.

In this note, we study the complementedness and the distributivity of upper semilattices of Kleene degrees assuming V = L. K denotes the upper semilattice of all Kleene degrees. We prove that if V = L, then some sub upper semilattices of K are non-complemented and some are non-distributive.

$\mathscr{K}$ 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.