Chains of P-points

TL;DR

Under the Continuum Hypothesis, the paper proves that increasing sequences of rapid P-points of length less than the continuum's successor are bounded above by a rapid P-point, simplifying key constructions in the field.

Contribution

It improves existing results by showing such sequences are bounded and simplifies the core definitions used in constructing long chains of P-points.

Findings

Sequences of rapid P-points are bounded under CH.

Equivalence of $oldsymbol{ extdelta}$-generic and weaker notions.

Simplification of the construction of long P-point chains.

Abstract

It is proved that the Continuum Hypothesis implies that any sequence of rapid P-points of length $<{\mathfrak c}^{+}$ which is increasing with respect to the Rudin-Keisler ordering is bounded above by a rapid P-point. This is an improvement of a result from [Kuzeljevi\'c, Raghavan: A long chain of P-points, arxiv:1607.07188 [math.LO]]. It is also proved that the notion of a $\delta$-generic sequence is equivalent to an apparently much weaker notion. This allows the central definition used in the construction in [Kuzeljevi\'c, Raghavan: A long chain of P-points, arxiv:1607.07188 [math.LO]] to be considerably simplified.