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

Logic