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	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