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