Advanced Course on Large Cardinals and Strong Logics

Abstract

The main purpose of mathematical logic is to devise formal languages in which one can reason about properties of mathematical structures, analyze them, classify them, and also establish relationships between di↵erent classes of structures. The most successful logical language is first order logic, which has a very streamlined and coherent model theory as well as a well-developed proof theory. Unfortunately, many natural mathematical concepts cannot be expressed in first order logic but need stronger logics, by which we mean extensions of first order logic by generalized quantifiers or infinitary operations. Examples of concepts that go beyond first order logic are some fundamental mathematical notions such as freeness of a group, separability of a space, completeness of an order, etc. When we extend first order logic we run immediately into set theoretical questions. On the one hand, the set of validities, and even the truth definition for those logics, model theoretic properties like the L¨owenheim-Skolem theorems, questions of compactness, interpolation theorems, etc., depend essentially on set theoretical assumptions such as strong infinitary combinatorial principles and the existence of large cardinals. On the other hand, some properties of natural strong logics are a source of set theoretical problems, well-known examples being Chang’s Conjecture and Stationary Reflection. Moreover, strong logics are relevant in the foundations of set theory. For instance they can be used to find interesting canonical inner models. There are some exciting new results in this direction.