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Proceedings of the American Mathematical Society

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

 

 

On the generic existence of special ultrafilters


Author: R. Michael Canjar
Journal: Proc. Amer. Math. Soc. 110 (1990), 233-241
MSC: Primary 03E05; Secondary 03E65, 04A20, 54A25
MathSciNet review: 993747
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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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DOI: https://doi.org/10.1090/S0002-9939-1990-0993747-3
Article copyright: © Copyright 1990 American Mathematical Society