The Combinatorics of Reasonable Ultrafilters

We are interested in generalizing part of the theory of ultrafilters on ω to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal. 0. Introduction A lot of knowledge has been accumulated on ultrafilters on ω. In the present paper we are interested in carrying out some of those arguments for ultrafilters on λ > א0, particularly if λ is strongly inaccessible. There is much work on normal ultrafilters, the parallel on ω are Ramsey ultrafilters. Now, every Ramsey ultrafilter on ω is a P -point but there are P -points of very different characters, e.g., P -point with no Ramsey ultrafilter below. Gitik [4] has investigated generalizations of P points for normal ultrafilters. Here we are interested in the dual direction, which up to recently I have not considered to be fruitful. In a long run, we are thinking of generalizing the following: (a) Consistently, some ultrafilters on ω are generated by < 20 many sets. (b) For a function f : ω −→ ω and ultrafilter D on ω, let D/f def = {A ⊆ ω : f(A) ∈ D}; it is an ultrafilter on ω (of course, we are interested in the cases when D and D/f are uniform, which in this case is the same as non-principal). By Blass and Shelah [1], consistently for any two non-principal ultrafilters D1, D2 on ω there are finite-to-one non-decreasing functions f1, f2 : ω −→ ω such that D1/f1 = D2/f2. (c) P -points are preserved by some forcing notions (see, e.g., [10, V], [8]) (d) For a significant family of forcing notions built according to the scheme of creatures of [8] we can consider an appropriate filter, i.e., if 〈pα : α < ω1〉 is ≤-increasing it may define an ultrafilter which is not necessarily generated by א1-sets, so we may ask on this. (e) Consistently, there is no P–point. The weakest demand we consider here for ultrafilters on λ is being weakly reasonable. What is a weakly reasonable ultrafilter on λ? It is a uniform ultrafilter on a regular cardinal λ which does not contain some club of λ and such that this property is preserved if we divide it by a non-decreasing f : λ −→ λ with unbounded range (see Definition 1.4 below). Date: July 2004. 1991 Mathematics Subject Classification. Primary 03E05; Secondary: 03E20. The author acknowledges support from the United States-Israel Binational Science Foundation (Grant no. 2002323). Publication 830. 1