We continue investigations of reasonable ultrafilters on uncountable cardinals defined in previous work by Shelah. We introduce stronger properties of ultrafilters 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 systems of size less than . We also show how ultrafilters generated by small systems can be killed by forcing notions which have enough reasonable completeness to be iterated with λ-supports.
"Reasonable Ultrafilters, Again." Notre Dame J. Formal Logic 52 (2) 113 - 147, 2011. https://doi.org/10.1215/00294527-1306154