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

我们研究了皮亚诺算术与不可辨别的理论（\(\textrm{PAI}\)）。\(\textrm{PAI}\)的模型的形式为\(({\mathcal {M}},I)\)，其中\({\mathcal {M}}\)是\(\ textrm{PA}\)，I是\({\mathcal {M}}\)上的无界有序集合，并且\(({\mathcal {M}},I)\)满足扩展归纳方案提到I 的公式。我们的主要结果是下面的定理 A 和 B。定理 A. 令 \({\mathcal {M}}\)为 任何基数 的\(\textrm{PA}\)的非标准模型 。\(\mathcal {M }\)具有对\(\textrm{PAI}\)模型的扩展，当且仅当\( {\mathcal {M}}\)具有归纳部分满足类。定理 A 产生以下推论，它提供了\(\textrm{PA}\)可数递归饱和模型的新特征：推论。 \ (\textrm{PA}\ )的可数模型\({\mathcal {M}} \)是递归饱和的，当且仅当\({\mathcal {M}}\)扩展为\(\textrm {PAI}\)。定理B.在算术语言中添加一元谓词I ( x )所获得的语言中存在一个句子\(\alpha \) ，使得给定任何非标准模型\({\mathcal {M}} \ )任何基数的(\textrm{PA}\)，\({\mathcal {M}}\)都扩展为\(\text {PAI}+\alpha \) iff \({\mathcal {M }}\)具有归纳的完全满意度等级。