On the Isbell problem

TL;DR

This paper constructs three models of ultrafilters on 9, showing the absence of basically generated ultrafilters in one and that all ultrafilters are Tukey top in the others, addressing the Isbell problem.

Contribution

It introduces new models of ultrafilters with specific Tukey type properties, providing solutions to the longstanding Isbell problem.

Findings

No basically generated ultrafilter in the first model.

All ultrafilters are Tukey top in the second and third models.

No -ultrafilter exists in any of the models.

Abstract

We present three models concerning Tukey types of ultrafilters on $\omega$. 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.