Independence and Large Cardinals

The independence results in arithmetic and set theory led to a proliferation of mathematical systems. One very general way to investigate the space of possible mathematical systems is under the relation of interpretability. Under this relation the space of possible mathematical systems forms an intricate hierarchy of increasingly strong systems. Large cardinal axioms provide a canonical means of climbing this hierarchy and they play a central role in comparing systems from conceptually distinct domains. This article is an introduction to independence, interpretability, large cardinals and their interrelations. Section 1 surveys the classic independence results in arithmetic and set theory. Section 2 introduces the interpretability hierarchy and describes some of its basic features. Section 3 introduces the notion of a large cardinal axiom and discusses some of the central examples. Section 4 brings together the previous themes by discussing the manner in which large cardinal axioms provide a canonical means for climbing the hierarchy of interpretability and serve as an intermediary in the comparison of systems from conceptually distinct domains. Section 5 briefly touches on some philosophical considerations.