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

2.
For each $x\in \mathbb{N}$, there exists exactly one ${x}^{\prime}\in \mathbb{N}$, called the successor^{} of $x$

3.
${x}^{\prime}\ne 0$ (0 is not the successor of any natural number)

4.
$x=y$ if and only if ${x}^{\prime}={y}^{\prime}$.

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
$x+1$  $=$  ${x}^{\prime}\mathit{\hspace{1em}}\text{for all}x\in \mathbb{N}$  
$x+{y}^{\prime}$  $=$  ${(x+y)}^{\prime}\mathit{\hspace{1em}}\text{for all}x,y\in \mathbb{N}$ 
Addition defined in this manner can then be proven to be both associative and commutative^{}.
Multiplication is
$x\cdot 1$  $=$  $x\mathit{\hspace{1em}}\text{for all}x\in \mathbb{N}$  
$x\cdot {y}^{\prime}$  $=$  $x\cdot y+x\mathit{\hspace{1em}}\text{for all}x,y\in \mathbb{N}$ 
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  20130322 12:32:42 
Last modified on  20130322 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 