Indiscernibles and satisfaction classes in arithmetic

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