### 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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