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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the generic existence of special ultrafilters
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by R. Michael Canjar
Proc. Amer. Math. Soc. 110 (1990), 233-241
DOI: https://doi.org/10.1090/S0002-9939-1990-0993747-3

Abstract:

We introduce the concept of the generic existence of $P$-point, $Q$-point, and selective ultrafilters, a concept which is somewhat stronger than the existence of these sorts of ultrafilters. We show that selective ultrafilters exist generically iff semiselectives do iff ${m_c} = c$, and we show that $Q$-point ultrafilters exist generically iff semi-$Q$-points do iff ${m_c} = d$, where $d$ is the minimal cardinality of a dominating family of functions and ${m_c}$ is the minimal cardinality of a cover of the real line by nowhere-dense sets. These results complement a result of Ketonen, that $P$-points exist generically iff $c = d$, and one of P. Nyikos and D. H. Fremlin, that saturated ultrafilters exist generically iff ${m_c} = c = {2^{ < c}}$.
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Bibliographic Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 110 (1990), 233-241
  • MSC: Primary 03E05; Secondary 03E65, 04A20, 54A25
  • DOI: https://doi.org/10.1090/S0002-9939-1990-0993747-3
  • MathSciNet review: 993747