On the generic existence of special ultrafilters

Canjar, R. Michael

doi:10.1090/S0002-9939-1990-0993747-3

On the generic existence of special ultrafilters
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Proc. Amer. Math. Soc. 110 (1990), 233-241 Request permission

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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