Lorenzo Galeotti, Amsterdam University College
Order types of models of arithmetic without induction
It is a well-known fact that non-standard models of Peano Arithmetic (PA) have order type N + Z · D, where D is a dense linear order. The question of which dense linear orders D can occur in such order types is non-trivial and widely studied. In this context Friedman asked the following question:
Are there consistent extensions of Peano Arithmetic T and T′ such that the class of order types of models of T and the class of order types of models of T′ differ?Friedman’s question is very complex and still wide open. In this talk we will go in the opposite direction and consider a version of Friedman’s question for syntactic fragments of PA. We will present results from a joint work with Benedikt Löwe on order types of non-standard models of syntactic subsystems of arithmetic obtained by restricting the language to subsets of the operations. We will put particular emphasis on models of syntactic subsystems of Peano Arithmetic obtained by dropping the schema of induction.