Peano arithmetic

Peano’s axioms are a definition of the set of natural numbers, denoted . From these axioms Peano arithmeticMathworldPlanetmathPlanetmath on natural numbersMathworldPlanetmath can be derived.

  1. 1.

    0 (0 is a natural number)

  2. 2.

    For each x, there exists exactly one x, called the successorMathworldPlanetmathPlanetmath of x

  3. 3.

    x0 (0 is not the successor of any natural number)

  4. 4.

    x=y if and only if x=y.

  5. 5.

    (axiom of induction) If M and 0M and xM implies xM, then M=.

The successor of x is sometimes denoted Sx instead of x. 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 additionPlanetmathPlanetmath and multiplication on natural numbers. Addition is defined as

x+1 = xfor all x
x+y = (x+y)for all x,y

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

Multiplication is

x1 = xfor all x
xy = xy+xfor all x,y

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