# Peano arithmetic

Peano’s axioms are a definition of the set of natural numbers, denoted $\mathbb{N}$. From these axioms Peano arithmetic on natural numbers can be derived.

1. 1.

$0\in\mathbb{N}$ (0 is a natural number)

2. 2.

For each $x\in\mathbb{N}$, there exists exactly one $x^{\prime}\in\mathbb{N}$, called the successor of $x$

3. 3.

$x^{\prime}\neq 0$ (0 is not the successor of any natural number)

4. 4.

$x=y$ if and only if $x^{\prime}=y^{\prime}$.

5. 5.

(axiom of induction) If $M\subseteq\mathbb{N}$ and $0\in M$ and $x\in M$ implies $x^{\prime}\in M$, then $M=\mathbb{N}$.

The successor of $x$ is sometimes denoted $Sx$ instead of $x^{\prime}$. We then have $1=S0$, $2=S1=SS0$, and so on.

Peano arithmetic consists of statements derived via these axioms. For instance, from these axioms we can define addition and multiplication on natural numbers. Addition is defined as

 $\displaystyle x+1$ $\displaystyle=$ $\displaystyle x^{\prime}\quad\text{for all }x\in\mathbb{N}$ $\displaystyle x+y^{\prime}$ $\displaystyle=$ $\displaystyle(x+y)^{\prime}\quad\text{for all }x,y\in\mathbb{N}$

Addition defined in this manner can then be proven to be both associative and commutative.

Multiplication is

 $\displaystyle x\cdot 1$ $\displaystyle=$ $\displaystyle x\quad\text{for all }x\in\mathbb{N}$ $\displaystyle x\cdot y^{\prime}$ $\displaystyle=$ $\displaystyle x\cdot y+x\quad\text{for all }x,y\in\mathbb{N}$

This definition of multiplication can also be proven to be both associative and commutative, and it can also be shown to be distributive over addition.

 Title Peano arithmetic Canonical name PeanoArithmetic Date of creation 2013-03-22 12:32:42 Last modified on 2013-03-22 12:32:42 Owner alozano (2414) Last modified by alozano (2414) Numerical id 8 Author alozano (2414) Entry type Axiom Classification msc 03F30 Related topic NaturalNumber Related topic PressburgerArithmetic Related topic ElementaryFunctionalArithmetic Related topic PeanoArithmeticFirstOrder Defines Peano’s axioms Defines successor Defines axiom of induction