Generalized Tukey reducibility between $\ddot{I}ƒ$-directed sets

Abstract

We introduce the pre-Tukey reducibility, a generalization of the Tukey reducibility between directed sets that works well in $\mathsf{ZF}$. We investigate the pre-Tukey reducibility between several $\ddot{I}ƒ$-directed sets under assumptions on sets of reals, which hold in the Solovay model and in $L(\mathbb{R})$ satisfying $\mathsf{AD}$.

19 pages