Some consistency strength analyses using higher core models (Q2726207)

This monograph is the author's Bonn dissertation. There are seven chapters. After an introductory chapter, chapter 2 reviews the theory of short core models. Chang's conjecture (CC) is taken up in chapter 3. The author shows that CC is equivalent to an apparently stronger version called \(\text{CC}^{\text{ club}}\), and he finds a combinatorial equivalent of \(\text{CC}^{\text{ club}}\). In chapter 4, a combinatorial principle closely related to CC, the Transversal Hypothesis (TH), is examined (\(\neg \text{TH}\) is a consequence of CC). The author considers a variant \(\text{TH}^{\text{stat}}\), whose negation is a consequence of \(\text{CC}^{\text{ club}}\). Using short core models, he determines a lower bound for the consistency strength of \(\neg \text{TH}^{\text{stat}}\) for cardinals beyond \(\aleph_1\). Chapter 5 is a review of the theorems and definitions for higher core models (core models up to a strong cardinal). In chapter 6 an estimate for the length of certain short iterations of mice is calculated. In the final chapter the author computes a new lower bound for the consistency strength of the existence of irregular ultrafilters.