Tutorial on Elementary Pairs of Models of Arithmetic

Almost all results in model theory of arithmetic are about elementary extensions. I will talk about an effort to systematize those results. If M and N are models of arithmetic and M ≺ N , how much information is there in Th((N,M)) in relation to Th(N)? In what situations is there a form of elimination of quantifiers for Th((N,M))? Is there a complete set of isomorphism invariants for such pairs? What is the diversity among theories of elementary pairs in general, and in situations where some of the isomorphism invariants are fixed? What is the right notion of genericity for elementary pairs? In the tutorial, I will introduce basic techniques that allow to approach such questions and I will talk about recent related results by Richard Kaye, Tin Lok Wong, Jim Schmerl, and myself. The results are specificifically about models of Peano Arithmetic, but the techniques are of general interest in model theory. Not all topics listed below will be covered in depth. Instead, I will concentrate on basic concepts and specific examples. Lecture 1 Indiscernible types and their applications. Models generated by indiscernibles. Gaps, interstices, and intersticial gaps. Rare and ubiquitous types. Lecture 2 The case of end extensions. Nonstandard standard systems. Classification of elementary cuts in recursively saturated models and in saturated models. Free cuts, generic cuts, and pseudostandard cuts. Lecture 3 The case of cofinal extensions. Greatest common initial segments. Bases. Finitely genereted extensions. Lattices of elementary substructures.