ordinal arithmetic

Ordinal arithmetic is the extensionPlanetmathPlanetmathPlanetmath of normal arithmeticPlanetmathPlanetmath to the transfinite ordinal numbers. The successor operation Sx (sometimes written x+1, although this notation risks confusion with the general definition of additionPlanetmathPlanetmath) is part of the definition of the ordinalsMathworldPlanetmathPlanetmath, and addition is naturally defined by recursion over this:

  • x+0=x

  • x+Sy=S(x+y)

  • x+α=supγ<α(x+γ) for limit ordinalMathworldPlanetmath α

If x and y are finite then x+y under this definition is just the usual sum, however when x and y become infiniteMathworldPlanetmath, there are differencesPlanetmathPlanetmath. In particular, ordinal addition is not commutativePlanetmathPlanetmathPlanetmathPlanetmath. For example,




Multiplication in turn is defined by iterated addition:

  • x0=0

  • xSy=xy+x

  • xα=supγ<α(xγ) for limit ordinal α

Once again this definition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to normal multiplication when x and y are finite, but is not commutative:




Both these functions are strongly increasing in the second argument and weakly increasing in the first argument. That is, if α<β then

  • γ+α<γ+β

  • γα<γβ

  • α+γβ+γ

  • αγβγ

Title ordinal arithmetic
Canonical name OrdinalArithmetic
Date of creation 2013-03-22 13:28:52
Last modified on 2013-03-22 13:28:52
Owner Henry (455)
Last modified by Henry (455)
Numerical id 7
Author Henry (455)
Entry type Topic
Classification msc 03E10
Related topic AdditivelyIndecomposable
Related topic CardinalArithmetic