properties of ordinal arithmetic

Let On be the class of ordinalsMathworldPlanetmathPlanetmath, and α,β,γ,δ𝐎𝐧. Then the following properties are satisfied:

  1. 1.

    (additive identity): α+0=0+α=α (proof (

  2. 2.

    (associativity of additionPlanetmathPlanetmath): α+(β+γ)=(α+β)+γ

  3. 3.

    (multiplicative identityPlanetmathPlanetmath): α1=1α=α

  4. 4.

    (multiplicative zero): α0=0α=0

  5. 5.

    (associativity of multiplication): α(βγ)=(αβ)γ

  6. 6.

    (left distributivity): α(β+γ)=αβ+αγ

  7. 7.

    (existence and uniqueness of subtraction): if αβ, then there is a unique γ such that α+γ=β

  8. 8.

    (existence and uniqueness of division): for any α,β with β0, there exists a unique pair of ordinals γ,δ such that α=βδ+γ and γ<β.

Conspicuously absent from the above list of properties are the commutativity laws, as well as right distributivity of multiplication over addition. Below are some counterexamples:

  • ω+11+ω=ω, for the former has a top element and the latter does not.

  • ω22ω, for the former is ω+ω, which consists an element α such that β<α for all β<ω, and the latter is 2sup{nn<ω}=sup{2nn<ω}=sup{nn<ω}, which is just ω, and which does not consist such an element α

  • (1+1)ω1ω+1ω, for the former is 2ω and the latter is ω2, and the rest of the follows from the previous counterexample.

All of the properties above can be proved using transfinite inductionMathworldPlanetmath. For a proof of the first property, please see this link (

For properties of the arithmeticPlanetmathPlanetmath regarding exponentiation of ordinals, please refer to this link (

Title properties of ordinal arithmetic
Canonical name PropertiesOfOrdinalArithmetic
Date of creation 2013-03-22 17:51:05
Last modified on 2013-03-22 17:51:05
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Result
Classification msc 03E10
Related topic OrdinalExponentiation