A long chain of P-points

TL;DR

This paper demonstrates that under the Continuum Hypothesis, there exists a long chain of P-points of length , extending previous work and answering a longstanding question in set theory.

Contribution

It introduces the concept of -generic sequences of P-points and proves their extension to -length chains under CH and Martin's Axiom.

Findings

Existence of -length chains of P-points under CH.

Extension of -generic sequences to longer chains.

Answer to an old question of Andreas Blass.

Abstract

The notion of a $\delta$-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis that for each $\delta < {\omega}_{2}$, any $\delta$-generic sequence of P-points can be extended to an ${\omega}_{2}$-generic sequence. This shows that the Continuum Hypothesis implies that there is a chain of P-points of length ${\mathfrak{c}}^{+}$ with respect to both Rudin-Keisler and Tukey reducibility. The proofs can be easily adapted to get such a chain of length ${\mathfrak{c}}^{+}$ under a more general hypothesis like Martin's Axiom. These results answer an old question of Andreas Blass.