Technical Report No. 2013-4 : Automorphisms of Models of Arithmetic (AU-CAS-MathStats)

The goal of this thesis is to exposit the key results of the subject, with an eye towards presenting two open problems related to the structure of the groups of automorphisms of a model of arithmetic. Section 2 will present preliminary results and definitions, discussing the axioms of Peano Arithmetic (PA), and deriving key theorems in the model theory of arithmetic. Section 3 will discuss recursive saturation, which is the key to building automorphisms of models of arithmetic. Essentially, recursive saturation is all about building models “just rich enough” to have many elements that are free to move under an automorphism. Finally, Section 4 will cover results at the cutting edge1 of the subject, and present two open problems.