Cascades, order, and ultrafilters

We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. This allows us to prove (ZFC) that the class of strict Jωω-ultrafilters, introduced by J.E. Baumgartner in [2], is empty. We translate the result to the language of <∞-sequences under an ultrafilter, investigated by C. Laflamme in [17], and we show that if there is an arbitrary long finite <∞-sequence under u, then u is at least a strict Jωω+1-ultrafilter.