Reasonable Ultrafilters, Again

We continue investigations of reasonable ultrafilters on uncount-able cardinals defined in Shelah [8]. We introduce stronger properties of ul-trafilters and we show that those properties may be handled in λ–support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal λ with generating system of size less than 2 λ. We also show how reasonable ultrafilters can be killed by forcing notions which have enough reasonable completeness to be iterated with λ–supports (and we show the appropriate preservation theorem). Reasonable ultrafilters were introduced in Shelah [8] in order to suggest a line of research that would repeat in some sense the beautiful theory created around the notion of P–points on ω. Most of the generalizations of P–points to uncountable cardinals in the literature goes into the direction of normal ultrafilters and large cardinals (see, e.g., Gitik [3]), but one may be interested in the opposite direction. If one wants to keep away from normal ultrafilters on λ, one may declare interest in ultrafilters which do not include all clubs and even demand that quotients by a closed unbounded subset of λ do not extend the club filter of λ. Such ultrafilters are called weakly reasonable ultrafilters, see 1.1, 1.2. But if we are interested in generalizing P–points, we have to consider also properties that would correspond to any countable family of members of the ultrafilter has a pseudo-intersection in the ultrafilter. The choice of the right property in the declared context of very non-normal ultrafilters is not clear, and the goal of the present paper is to show that the very reasonable ultrafilters suggested in Shelah [8] (see Definition 1.3 here) are very reasonable indeed, that is we may prove interesting theorems on them. In the first section we recall some of the concepts and results presented in Shelah [8] and we introduce strong properties of generating systems (super and strong reasonability, see Definitions 1.11, 1.12) and we show that there may exist super reasonable systems generating ultrafilters (Propositions 1.15, 1.16). In the next section we remind from [6] some properties of forcing notions relevant for λ–support iterations. We also improve in some sense a result of [6] and we show a preservation theorem for nice double a–bounding property (Theorem 2.12).