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Cascades, order, and ultrafilters. (English) Zbl 1354.03061

Summary: 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 \(J_{\omega^\omega}\)-ultrafilters, introduced by J. E. Baumgartner [J. Symb. Log. 60, No. 2, 624–639 (1995; Zbl 0834.04005)], is empty. We translate the result to the language of \(<_\infty\)-sequences under an ultrafilter, investigated by C. Laflamme [J. Symb. Log. 61, No. 3, 920–927 (1996; Zbl 0871.04004)], and we show that if there is an arbitrary long finite \(<_\infty\)-sequence under \(u\), then \(u\) is at least a strict \(J_{\omega^{\omega+1}}\)-ultrafilter.

MSC:

03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
03E20 Other classical set theory (including functions, relations, and set algebra)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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References:

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