Cascades, Order and Ultrafilters

TL;DR

This paper explores the relationship between cascades and ultrafilters, proving the non-existence of strict J_{ω^ω}-ultrafilters in ZFC and relating ultrafilter sequences to their strictness levels.

Contribution

It establishes the emptiness of strict J_{ω^ω}-ultrafilters in ZFC and connects ultrafilter sequence length to their strictness classification.

Findings

Strict J_{ω^ω}-ultrafilters do not exist in ZFC.

Long finite sequences under an ultrafilter imply at least strict J_{ω^{ω+1}}-ultrafilter.

The paper links ultrafilter behavior with ordinal sequence properties.

Abstract

We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. Using that relation we prove (ZFC) that the class of strict $J_{\omega^\omega}$-ultrafilters, introduced by J. E. Baumgartner in \textit{Ultrafilters on $\omega$}, is empty. We translate the result to the language of $<_\infty$-sequences under an ultrafilter, investigated by C. Laflamme in \textit{A few special ordinal ultrafilters}, to show that if there is an arbitrary long finite $<_\infty$-sequence under $u$ than $u$ is at least strict $J_{\omega^{\omega+1}}$- ultrafilter.