# Peano arithmetic

**Peano arithmetic** An axiomatic theory, based on first-order logic, about the natural numbers. Its axioms are first-order statements about the simple arithmetic operations of successor *n*+1, addition *n*+*m*, and multiplication *n**.**m*, together with the principle of induction. A great variety of facts in mathematics and computer science can be reduced to, or can be coded as, statements about natural numbers, and Peano arithmetic is a strong enough theory to express and prove formally the majority of basic results. However, Gödel's incompleteness theorems show that there are first-order statements that are true of the natural numbers but are not provable in Peano arithmetic, or its extensions. Although many algebras satisfy the axioms of Peano arithmetic, there is only one computable algebra that satisfies the axioms and that is the so-called *standard model of Peano arithmetic*, namely the algebra ({0,1,2, …} | 0, *n*+1, *n*+*m*, *n**.**m*)

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**Peano arithmetic**