# NCF, P-points, weak P-points, and cardinalities

The post is a bit long, but all the questions are similar or concern the same topic.

Let $$\omega^*=\beta\omega\setminus\omega$$. A well-known topological definition of a P-point (on $$\omega$$) is as follows: a point $$x\in\omega^*$$ is a P-point if every intersection of countably many open neighborhoods of $$x$$ contains an open neighborhood of $$x$$. Similarly, a point $$x\in\omega^*$$ is a weak P-point if $$x$$ is not in the closure of any countable subset not containing $$x$$. We have also an equivalent definition of P-points in terms of functions $$\omega\to\omega$$: an ultrafilter $$x\in\omega^*$$ is a P-point if for every function $$f\colon\omega\to\omega$$ there is $$A\in x$$ such that $$f\restriction A$$ is either constant or finite-to-one. Since every P-point is a weak P-point, my first question is as follows:

Question 1. Does there exist a similar characterization of weak P-points (i.e. in terms of functions $$\omega\to\omega$$)?

Kunen proved in ZFC that we always have $$2^{\mathfrak{c}}$$ weak P-points and at least $$\mathfrak{c}$$ incomparable in the sense of Rudin-Keisler weak P-points. My next questions concern the total number of (incomparable/incompatible) P-points provided that there is at least one P-point.

Question 2. Assume that a P-point exists. (a) Does there exist another (non-isomorphic or incomparable) P-point? (b) Do there exist $$2^{\mathfrak{c}}$$ different P-points? (c) Do there exist $$\mathfrak{c}$$ incomparable P-points?

(It is easy to see that there exist $$\mathfrak{c}$$ isomorphic P-points.)

Question 3. Assume that there exist $$2^{\mathfrak{c}}$$ many P-points. (a) Do we have then $$2^{\mathfrak{c}}$$ (or at least $$\mathfrak{c})$$ incomparable P-points? (b) Do there exist $$2^{\mathfrak{c}}$$ (or at least $$\mathfrak{c}$$, or even $$2$$) incompatible P-points?

Let us say that two ultrafilters $$U,V\in\omega^*$$ are near coherent if there exists a finite-to-one function $$f\colon\omega\to\omega$$ such that $$f(U)=f(V)$$. The near coherence is an equivalence relation, so we can count the number of the equivalence classes. The Near Coherence of Filters principle (NCF in short) states that there exists only one equivalence class. Blass and Shelah constructed a model of set theory in which the NCF holds (it is now also known to hold in the Miller model). On the other hand, Banakh and Blass proved that either we have finitely many equivalence classes or $$2^{\mathfrak{c}}$$ (the latter holds e.g. in each model where $$\mathfrak{u}\ge\mathfrak{d}$$, so e.g. under CH). The next question is in the same spirit as Questions 2 and 3.

Question 4. Assume that we have $$2^{\mathfrak{c}}$$ many near coherence classes. Does there exist $$\mathfrak{c}$$ (or $$2^{\mathfrak{c}}$$) many incompatible in the sense of Rudin-Keisler ordering weak P-points?

It is believed (but not proved so far) that there exists a model with exactly $$2$$ near coherence classes and for every $$n>2$$ there is no model with $$n$$ classes. Also, the NCF implies the existence of a P-point. Thus, my next (and last) question is the following.

Question 5. Assume there are exactly 2 classes of near coherence. Does there exist any P-point? If yes, then are there two that are not compatible?

Thank you very much for the help!

An answer to question 2: Shelah constructed a model with exactly one $$P$$-point (up to isomorphism). In fact, the one $$P$$-point is a selective ultrafilter. You can find the construction in section XVIII.4 of Proper and Improper Forcing.
An answer to question 5: If there are exactly $$2$$ classes of near coherence, then $$\mathfrak{u} < \mathfrak{d}$$. (You say this in your post, just before Question 4.) In other words, there is a non-principal ultrafilter generated by fewer than $$\mathfrak{d}$$ sets. Ketonen proved that any ultrafilter generated by fewer than $$\mathfrak{d}$$ sets is a (non-selective) $$P$$-point. See
J. Ketonen, "On the existence of $$P$$-points in the Stone-Cech compactification of the integers," Fundamenta Mathematicae 92 (1976), pp. 91-94. (link)
• Note that $\mathfrak{u}=\mathfrak{d}=\aleph_1$ in this model, so you have $2^{\mathfrak{c}}$ near coherence classes as well. You didn't ask about this specifically, but it seemed relevant to the spirit of your question. – Todd Eisworth Mar 20 '20 at 12:34
• For the second part of question 5, let me point out that finitely many equivalence classes implies there are non-isomorphic $P$-points. We already know it gives us $P$-points that are not selective. But an ultrafilter is RK-minimal iff it is selective, so there are $P$-points with ultrafilters strictly RK-below them. But anything RK-below a $P$-point is also a $P$-point. – Will Brian Mar 20 '20 at 13:04