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This question is motivated by discussion surrounding this MO question.

An ultrafilter $U$ on $\omega$ is a simple $P_{\aleph_2}$-point if it is generated by a sequence $\langle X_\alpha:\alpha<\omega_2\rangle$ such that

$$\alpha<\beta\Longrightarrow |X_\beta\setminus X_\alpha|<\aleph_0,$$

that is, generated by an $\omega_2$ sequence of subsets of $\omega$ that is decreasing modulo the ideal of finite sets.

Question

Is the existence of a simple $P_{\aleph_2}$-point consistent with $\mathfrak{b}=\aleph_1$?

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    $\begingroup$ Did you mean $\beta < \alpha$? $\endgroup$ Feb 6, 2020 at 18:34
  • $\begingroup$ Fixed it. Thanks! $\endgroup$ Feb 7, 2020 at 4:34

1 Answer 1

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The answer is "yes". Alan Dow pointed out in correspondence that the construction of Blass and Shelah referenced below can be modified to yield such a model. The particular model arises by using finite support to add an $\omega_1$-sequence of Cohen reals, followed by a finite support iteration adding the generating set for the $P_{\aleph_2}$-point using a variant of Mathias forcing. The salient point of the construction is that the Mathias-type forcing (really Mathias forcing with respect to a carefully chosen ultrafilter at each stage) does not add a real dominating the Cohen reals, and so $\mathfrak{b}=\aleph_1$ in the extension.

Referring to the discussion on the question of Banakh that motivated our question, this shows there is a model in which Banakh's cardinal $\kappa$ is strictly greater than $\mathfrak{b}$, as the $P_{\aleph_2}$-point is an ultrafilter $U$ with $\tau(U)=\aleph_2$.

Blass, Andreas; Shelah, Saharon, Ultrafilters with small generating sets, Isr. J. Math. 65, No. 3, 259-271 (1989). ZBL0681.03033.

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  • $\begingroup$ Great! I only now noticed this your very helpful post. Thank you. $\endgroup$ Feb 18, 2020 at 3:12
  • $\begingroup$ Is it true that $\aleph_2=\mathfrak c$ (or at least $\aleph_2$ can be equal to $\mathfrak c$) in your model? $\endgroup$ Feb 18, 2020 at 3:21
  • $\begingroup$ What is the value of $\mathrm{cov}(\mathcal M)$ in your model? Can it be equal to $\aleph_1$? This would help to shed light to another my question: mathoverflow.net/q/352976/61536 $\endgroup$ Feb 18, 2020 at 6:50

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