Cascades, Order and Ultrafilters

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.

12 pages

References in corpus (1)

• On what I do not understand (and have something to say): Part I

Cited by in corpus (1)

• Ordinal ultrafilters versus P-hierarchy