On the Isbell problem

Abstract

We present three models concerning Tukey types of ultrafilters on $\ddot{I}‰$. The first model is built via a countable support iteration, and we show there is no basically generated ultrafilter in such model. The second and third models are built upon different and novel techniques, and in such models all ultrafilters are Tukey top, thus providing an answer to the Isbell problem. In all models there is no $\mathsf{nwd}$-ultrafilter.

Changes relative to v1: Section 6 has been reorganized into three new sections, each one developing one specific topic. The notion of $\ddot{I}‚$-proper forcing has been renamed to Cohen proper forcing. The preservation theorem for iterations of Cohen proper forcing has been reformulated in a simpler way. Other minor corrections have been done. Changes relative to v2: only minor modifications