Indiscernibles and satisfaction classes in arithmetic

Abstract 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 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+ PAI + α iff {M} M has a inductive full satisfaction class.