Chang’s Conjecture, generic elementary embeddings and inner models for huge cardinals

These lectures summarize and organize the material appearing in: Smoke and Mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals. • Much of the early history of logic in general and model theory in particular was tied up with understanding the expressive power of first and second order logic (and their variants). • One distinguishing feature of first order logic is the Downwards Lowenheim-Skolem Theorem. • Tremendous effort was put into generalizing the downwards Lowenheim-Skolem theorem so that the elementary substructure had some second order properties • The coarsest second order properties had to do with cardinality; in this discussion we consider various more subtle second order properties. Among them are being correct for the non-stationary ideal. Let L be a countable language with a distinguished unary predicate R. Then is said to have type (κ, λ) if and only iff |A| = κ and |R A | = λ.