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Nonregular ultrafilters on \(\omega _{2}\). (English) Zbl 1243.03064

A uniform ultrafilter \(U\) on \(\omega_2\) is said to be fully nonregular if for any size-\(\aleph_2\) subset \(X\) of \(U\) there is a size-\(\aleph_1\) subset \(Y\) of \(X\) such that \(\bigcap Y \not= \emptyset\). It is shown that if there is such an ultrafilter then (a) there is an inner model with a cardinal \(\nu\) of Mitchell order \(\nu^+\), and (b) if \(| {}^{\omega_2} \omega_1 / U| = \aleph_2\) then there are mice extenders with multiple generators.

MSC:

03E35 Consistency and independence results
03E55 Large cardinals
03E05 Other combinatorial set theory
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