Reasonable ultrafilters
by
Andrzej Roslanowski
University of Nebraska at Omaha
Coauthors: Saharon Shelah

Reasonable ultrafilters on uncountable cardinals were introduced by Shelah ([Sh:830]) in order to suggest a line of research that would in some sense repeat the beautiful theory created around the notion of P-points on w. The definition of reasonable ultrafilters involves two conditions. The first one, so called the weak reasonability of an ultrafilter, is a way to guarantee that we are not entering the realm of large cardinals: the considered ultrafilter is required to be very non-normal.

Definition:    Let D be a uniform ultrafilter on a regular uncountable cardinal l. We say that D is weakly reasonable, if for every increasing continuous sequence 〈d_x:x < l〉 ⊆ l there is a club C^* of l such that

∪{[dx, dx+1):x ∈ C*} ∉ D.

The second part of the definition of reasonable ultrafilters is directly related to generalizing P-points to the context of weakly reasonable ultrafilters on an uncountable cardinal l. To carry out this process we have to be somewhat creative in re-interpreting the property that any countable family of members of the ultrafilter has a pseudo-intersection in the ultrafilter.

We consider sequences r=〈(a_x, d_x):x < l〉 such that 〈a_x:x < l〉 is an increasing continuous sequence of ordinals below l and d_x is an ultrafilter on the interval [a_x, a_x+1). For each such sequence r we look at the family of subsets of l which are eventually large in every interval [a_x, a_x+1) , that is we consider the set

fil(r)={A ⊆ l:(∃z < l)(∀x > z)(A∩[ax, ax+1) ∈ dx)}.

(The set fil(r) is a filter on l.) There is a natural quasi-ord er on sequences r as above: we say that r ≤ ^* s if and only if fil(r) ⊆ fil(s). Now the demand generalizing P-pointness may be phrased for an ultrafilter D on l as follows:

(*) there is a ( < l⁺)-directed (with respect to ≤ ^*) family H such that D=∪{fil(r):r ∈ H}.

In the talk we will present a review of the main results concerning reasonable ultrafilters and their relatives which are included in two papers by Roslanowski and Shelah ([RoSh:889] and [RoSh:890]).

Paper reference: arXiv:math.LO/0607218, arXiv:math.LO/0605067

Date received: March 19, 2008