Hausdorff ultrafilters I know that Ramsey ultrafilters are Hausdorff ($\mathcal{U}$ is Hausdorff iff for every $f,g:\mathbb{N}\rightarrow\mathbb{N}$ $f(\mathcal{U})=g(\mathcal{U})$ then $f\cong_\mathcal{U} g$ $\;$). So if we assume the continuum hypothesis then there exist Hausdorff ultrafilters. I'm wondering this: if we assume continuum hypothesis then there exists a Hausdorff ultrafilter not Ramsey?
 A: Under the continuum hypothesis, or even Martin's Axiom, there are Hausdorff ultrafilters that are not selective.  The following is a good survey by Fremlin of many related matters:
http://www.essex.ac.uk/maths/people/fremlin/n09102.pdf
The construction, under CH, of such ultrafilters goes back to Andreas Blass, in the 1970s.
The harder question of whether the existence of Hausdorff ulrafilters can be proved in ZFC was settled not many years ago.  The answer is no.  Shelah is one of the authors.
A: The paper“On the Existence of Nonprincipal Arithmetical Ultrafilters on ω” was appeared in ACTA MATHEMATICA SINICA CHINESE SERIES Mar.2006,49(2): 283-288.
In this paper it is proved that the statement (Q) implies the existence of nonprincipal arithmetical (i.e. Hausdorff ) ultrafilters on ω.
Statement (Q): If B ⊂ P(ω) has the sfip (strong finite intersection property) and |B| < 2^ω, then there is a Q-point q ⊃ B.
Statement (Q) is strongly weaker than Statement (R).
Statement (R): If B ⊂ P(ω) has the sfip (strong finite intersection property) and |B| < 2^ω, then there is a  Ramsey ultrafilter r ⊃ B.
  Fangting Wang 
A: On the Existence of Nonprincipal Arithmetical
Ultrafilters on ω
Fangting Wang
Department of Mathematics
University of Science and Technology of China
Hefei, Anhui, 230026 P.R.China
Email: ftwang@ustc.edu.cn
January 8, 2013
Abstract
Each nonprincipal arithmetical ultrafilter p ∈ βω−ω is associated with
a simple arithmetical model Np= {f(p) : f ∈ωω} ⊃ N, and a positive real
number is just an equivalence class of finite fractions made by arithmetical
ultrafilters (instead of natural numbers), just like the ancient Greek’s idea.
It is well known that MAcountableimplies the statement (Q):
1
(Q)
If B ⊂ P(ω) has the sfip (strong finite intersection property) and
|B| < 2ω, then there is a Q-point q ⊃ B.
In this paper we will prove that the statement (Q) implies our Hypoth-
esis:
There exist nonprincipal arithmetical ultrafilters on ω.
Key words:
the Stone-ˇCech compactification on ω, Q-point,
arithmetical
ultrafilter, arithmetical model
MR(2000): 03E35, 03E65, 54D80, 54D35, 03C62
1
Introduction
βω, the Stone-ˇCech compactification of the discrete space ω, consists of all
ultrafilters on ω. For p ∈ βω and f ∈
ωω, the image of p in βω under the Stone
extention f : βω → βω is given by
f(p) = {a ⊂ ω : f−1[a] ∈ p}.
For f,g ∈
ωω and p ∈ βω, f =pg means that
{n ∈ ω : f(n) = g(n)} ∈ p.
An ultrafilter p ∈ βω is called arithmetical [1], if for all f,g ∈
ωω,
(1)
f(p) = g(p) ↔ f =pg.
2
(The nonprincipal arithmetical ultrafilter was called ∗-point in [2].)
For all f,g ∈
ωω and all ultrafilters p ∈ βω we always have
(2)
f =pg → f(p) = g(p),
but the converse of (2) may not be true [2]. Because of (2), an ultrafilter p is
arithmetical iff for all f,g ∈
ωω,
(3)
f(p) = g(p) → f =pg.
We are interested in this class of ultrafilters since each arithmetical ulrtrafilter
p is associated with a simple arithmetical model Np:
Np= {f(p) : f ∈
ωω}.
In Npthe operations +,· and h(∈
ωω) are defined naturally:
f(p) + g(p) = (f + g)(p),
f(p) · g(p) = (f · g)(p),
h(f(p)) = (h ◦ f)(p).
Npis a structure of the language L = {0,+,·} ∪ {f : f ∈
ωω}. We have the
Transfer between Npand N (the standard arithmetical model):
Transfer For any L∪ {p} -sentence ϕ(p),
Np|= ϕ(p) ↔ {n ∈ ω : N |= ϕ(n)} ∈ p.
3
Specially, for any L-sentence ϕ,
Np|= ϕ ↔ N |= ϕ.
The proof of the Transfer is completed by a simple induction on the length of
ϕ(p). The first step is the property (1) which the arithmetical ultrafilter p has.
Is there a point in βω − ω which is arithmetical? Our Hypothesis (AR) is:
(AR)
There exist nonprincipal arithmetical ultrafilters on ω.
A point q ∈ βω − ω is called a Q-point, if for every partition (ai)i∈ωof ω into
pairwise disjoint finite subsets there exists b ∈ q with |b ∩ ai| ≤ 1 for all i ∈ ω.
We write sfip for “strong finite intersection property”, and let (Q) be the
statement:
(Q)
If B ⊂ P(ω) has the sfip and |B| < 2ω, then there is a Q-point q ⊃ B.
Then we have:
Main Theorem The statement (Q) implies the Hypothesis (AR).
Before we prove the main theorem, we list some well-known results about the
existence of Q-points:
1◦MAcountableimplies the statement (Q), and the word “Q-points” in the
statement (Q) can be replaced by “selective (minimal, or Ramsay) ultrafilters”
4
(cf.[3], Th.2).
2◦If there exists a dominant family (inωω) of cardinality ω1, then there exists
a Q-point [4].
3◦the assumption of the existence of Q-point is weaker than MAcountable, since
MAcountableimplies that the cardinality of each dominant family (inωω) must
be 2ω[5].
2
Lemmas
The following lemma 1 improves the proposition 2 in [2].
Lemma 1
For any f,g ∈
ωω and p ∈ βω , if f(p) = g(p) and (f → g)p,
then f =pg, where (f → g)pmeans that
∃a ∈ p ∀x,y ∈ a (f(x) = f(y) → g(x) = g(y)).
Proof Assume f(p) = g(p) and (f → g)p. It suffices to show that both f  h(ni). Hence
∀ni∈ f[ω]∃k ∈ ω(hk(ni) / ∈ f[ω]).
Now let
A = {ni∈ f[ω] : h(ni) ∈ g[ω] − f[ω]},
B = {ni∈ f[ω] − A : the least k such that hk(ni) / ∈ f[ω] is even},
C
= f[ω] − A − B.
The remaining part is the same as in the case f 
