Indiscernibles and satisfaction classes in arithmetic

Abstract

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.