Reasonable ultrafilters, again

by Roslanowski and Shelah. [RoSh:890]
Notre Dame J Formal Logic, 2011
We continue investigations of reasonable ultrafilters / on uncountable cardinals defined in [Sh:830]. We introduce stronger properties of ultrafilters and we show that those properties may be handled in lambda --support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal lambda with generating system of size less than 2^lambda . We also show how reasonable ultrafilters can be killed by forcing notions which have enough reasonable completeness to be iterated with lambda --supports (and we show the appropriate preservation theorem).

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