Towers and Ultrafilters(joint work J. Brendle and B. Farkas)

FF UK

Department of Logic

Jonathan VernerDisclaimerFF UK

Department of Logic

The views presented here are my own and do not necessarily reflect
those of my coauthors.

Motivation

FF UK

Department of Logic

Definition A tower is a descending sequence in ([ω]ω , ⊆∗ ) withno lower bound.
Observation Assume CH. Then every P-point is generated by atower.
Question▶ Can the assumption of being a P-point be weakened/dropped?
▶ Can CH be weakened/dropped?

Putting towers into ultrafilters

FF UK

Department of Logic

Observation If an ultrafilter is generated by a tower, then it is aP-point.
Question Can every ultrafilter contain a tower?

A naive attempt

FF UK

Department of Logic

Putting towers into ultrafilters

A naive attempt▶ Enumerate [ω]ω as {Xα : α < c}▶ start with some T0 ∈ U and recursively construct ⊆∗ -descending

Tα ∈ U; at step α let Tα+1 be either Tα ∩ Xα or Tα \ Xα ,depending on which is in U

▶ What to do at limits?
▶ It gets even worse: It can happen, that U contains a tower but at

the same time has a basis such that no infinite sequence in thebasis has a pseudointersection in U. (For all we know, all ultrafilters
can have such a basis!)▶ Under CH it fails (Kunen, van Mill, Mills)!

RTFM

FF UK

Department of Logic

Balcar, Frankiewicz, Mills: More on nowhere dense closed P-sets, Bull.
Acad. Polon. Sci. Sér. Sci. Math, 28 (1980)▶ the paper is published in an obscure journal, not available online▶ need to know topology

Theorem It is consistent that ω ∗ can be covered by nowhere
dense closed P-sets.
Translation It is consistent that every ultrafilter contains a tower.

Pull out the big guns

FF UK

Department of Logic(an alternative proof)

Observation There always is some ultrafilter, which contains atower!
Make all ultrafilters look alike!

Definition (Blass) Two ultrafilters V0 , V1 are nearly coherent if
there is an ultrafilter U which is RB-below both of them.
Theorem (Blass) It is consistent that every two nonprincipal
ultrafilters are nearly coherent.

Transfering towers

FF UK

Department of Logic(under NCF)

Observation If U contains a tower than all ultrafilters RB-above
U also contain towers.
Theorem If a tower T of length b generates a nonmeager filter
and V is an ultrafilter containing T , then any ultrafilter RB-below
contains a tower.
Theorem (Solomon,Simon) If b < d then there is a tower oflength b which generates a non-meager filter.
Note Under NCF every tower of length c is meager! (It is notclear, whether there are towers of length c.)Weakening CHFF UK

Department of Logic

P-points without towers

Question Assume MA. Does every P-point contain a tower?
Question Does d = c imply that every P-point contains a tower?
Theorem Let ω < λ ≤ κ be regular cardinals. Then it isconsistent that▶ t = λ ≤ κ = c; and▶ There is a selective ultrafilter which contains no tower.
In particular, it is consistent that d = c and there is a P-point notcontaining a tower.

Constructing P-points without towers

FF UK

Department of Logic▶ Assume t = c and ♢(Sc≥ω1 )▶ Recursively construct the ultrafilter in c many steps▶ Guarantee the following condition: there is a family {Xα : α < c},such that if X ∈ U then Xα ⊆ X for all but boundedly many α.
▶ To get t < c, use a trick of Shelah:Start with GCH and let B = (λ<λ , ⊇).
▶ Force t = c = κ, then force the diamond.
▶ Finally, force with BV .
▶ This does not add reals and adds a tower of length λ thus making
t = λ.
▶ Finally, show that BV does not add long towers into the ultrafilter.
▶