Indiscernibles and satisfaction classes in arithmetic

Abstract

We investigate the theory Peano Arithmetic with Indiscernibles (\(\textrm{PAI}\)). Models of \(\textrm{PAI}\) are of the form \(({\mathcal {M}},I)\), where \({\mathcal {M}}\) is a model of \(\textrm{PA}\), I is an unbounded set of order indiscernibles over \({\mathcal {M}}\), and \(({\mathcal {M}},I)\) satisfies the extended induction scheme for formulae mentioning I. Our main results are Theorems A and B following. Theorem A. Let \({\mathcal {M}}\) be a nonstandard model of \(\textrm{PA}\) of any cardinality. \(\mathcal {M }\) has an expansion to a model of \(\textrm{PAI}\) iff \( {\mathcal {M}}\) has an inductive partial satisfaction class. Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of \(\textrm{PA}\): Corollary. A countable model \({\mathcal {M}}\) of \(\textrm{PA}\) is recursively saturated iff \({\mathcal {M}}\) has an expansion to a model of \(\textrm{PAI}\). Theorem B. There is a sentence \(\alpha \) in the language obtained by adding a unary predicate I(x) to the language of arithmetic such that given any nonstandard model \({\mathcal {M}}\) of \(\textrm{PA}\) of any cardinality, \({\mathcal {M}}\) has an expansion to a model of \(\text {PAI}+\alpha \) iff \({\mathcal {M}}\) has a inductive full satisfaction class.