Non-closure of the image model and absence of fixed points (Q1069933)

The existence of elementary embeddings j:V\(\to M\) of the universe into some inner model with suitable closure properties leads to various large cardinal axioms. The author shows that on the other hand also the existence of elementary embeddings violating certain closure conditions (e.g. \(^{\omega}M\subseteq M\) and \(^{\omega_ 1}M\varsubsetneq M)\) also implies the existence of large cardinals (in this case: there is an inner model with uncountably many measurable cardinals). Examples are given of ultrafilters which give rise to elementary embeddings of the latter kind. In the second part the author shows that also the absence of some fixed points of elementary embeddings is related to large cardinal assumptions.