Indiscernibles and satisfaction classes in arithmetic

We investigate the theory Peano Arithmetic with Indiscernibles ( PAI PAI ). Models of PAI PAI are of the form (M,I) ( M , I ) , where M M is a model of PA PA , I is an unbounded set of order indiscernibles over M M , and (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 M M be a nonstandard model of PA PA of any cardinality . M M has an expansion to a model of PAI PAI iff 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 PA PA : Corollary. A countable model M M of PA PA is recursively saturated iff M M has an expansion to a model of PAI PAI . Theorem B. There is a sentence α α in the language obtained by adding a unary predicate I ( x ) to the language of arithmetic such that given any nonstandard model M M of PA PA of any cardinality , M M has an expansion to a model of PAI+α PAI + α iff M M has a inductive full satisfaction class.