Indiscernibles and satisfaction classes in arithmetic

TL;DR

This paper explores models of Peano Arithmetic extended with indiscernibles and satisfaction classes, establishing new characterizations of models using these concepts and linking them to satisfaction classes and saturation.

Contribution

It introduces the theory PAI with indiscernibles, providing new characterizations of models of PA via satisfaction classes and recursive saturation.

Findings

Models of PAI correspond to models with inductive partial satisfaction classes

Countable recursively saturated models of PA can be expanded to models of PAI

Existence of a specific sentence s characterizes models with inductive full satisfaction classes

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.