UC Irvine

The purpose of this thesis is to use the tools of Inner Model Theory to the study of notions relative to generic embeddings induced by ideals. We seek to apply the Core Model Induction technique to obtain lower bounds in consistency strength for a specific Stationary Catching principle called StatCatch*(I), related to the saturation of an ideal I of omega2. This principle involves the central notion of self-genericity in its formulation, introduced by Foreman, Magidor and Shelah. In particular, we show that assuming StatCatch*(I) (plus some additional hypothesis in the universe), we can obtain, for every finite number n, an inner model with n Woodin Cardinals.