Journal of Symbolic Logic

Saturating Ultrafilters on N

D. H. Fremlin and P. J. Nyikos

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We discuss saturating ultrafilters on $\mathbf{N}$, relating them to other types of nonprincipal ultrafilter. (a) There is an $(\omega,\mathfrak{c})$-saturating ultrafilter on $\mathbf{N} \operatorname{iff} 2^\lambda \leq \mathfrak{c}$ for every $\lambda < \mathfrak{c}$ and there is no cover of $\mathbf{R}$ by fewer than $\mathfrak{c}$ nowhere dense sets. (b) Assume Martin's axiom. Then, for any cardinal $\kappa$, a nonprincipal ultrafilter on $\mathbf{N}$ is $(\omega,\kappa)$-saturating iff it is almost $\kappa$-good. In particular, (i) $p(\kappa)$-point ultrafilters are $(\omega,\kappa)$-saturating, and (ii) the set of $(\omega,\kappa)$-saturating ultrafilters is invariant under homeomorphisms of $\beta\mathbf{N\backslash N}$. (c) It is relatively consistent with ZFC to suppose that there is a Ramsey $p(\mathfrak{c})$-point ultrafilter which is not $(\omega,\mathfrak{c})$-saturating.

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J. Symbolic Logic, Volume 54, Issue 3 (1989), 708-718.

First available in Project Euclid: 6 July 2007

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Fremlin, D. H.; Nyikos, P. J. Saturating Ultrafilters on N. J. Symbolic Logic 54 (1989), no. 3, 708--718.

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