On order-types of models of arithmetic

Synopsis In this thesis we study a range of questions related to the order structure of models of first-order Peano Arithmetic. In Chapter 1 we give necessary definitions and describe the current state of the subject in the literature survey. In Chapter 2 we study first properties of order-types of models of PA, give examples and place first restrictions on what the order-type of a model of PA can be. In Chapter 3 we study models generated by indiscernibles and prove that the model generated by a set of indiscernibles ordered as a dense linear order (C, <) is order-embeddable into C <Q. In Chapter 4 we study a wide variety of questions associated with inter-pretability in a model of PA. In section 4.1 we prove that there is only one dense linear order interpreted in Ω |= PA, namely Q(Ω). In section 4.2 we express the order-type of all inner models in terms of the (<, ·)-structure of the outer model. In section 4.3 we introduce and study the notion of self-similarity. Section 4.4 briefly studies connections between models of PA and models of ZFC and their mutual interpretability. Chapter 5 introduces a class of canonical orders of models of PA and makes the first attempt to study it. In Chapter 6 we prove that every model of cardinality less than λ has 2 λ order-types of elementary end-extensions of cardinality λ, that there are 2 ω 1 ω 1-like models of PA and that (assuming ♦) there are 2 ω 1 ω 1-like models of PA whose multiplicative reducts are pairwise non-isomorphic. Sections 7.1 and 7.3 present our first attempt to solve Friedman's problem in the resplendent case. Section 7.2 connects this theme with the notion of arithmetic saturation. Also, we obtain a consequence of arithmetic saturation for automorphism groups. The thesis concludes with a list of questions intended to guide and inspire future research.