von Neumann ordinal

The is a method of defining ordinalsMathworldPlanetmathPlanetmath in set theoryMathworldPlanetmath.

The von Neumann ordinal α is defined to be the well-ordered set containing the von Neumann ordinals which precede α. The set of finite von Neumann ordinals is known as the von Neumann integers. Every well-ordered set is isomorphic to a von Neumann ordinal.

They can be constructed by transfinite recursion as follows:

  • The empty setMathworldPlanetmath is 0.

  • Given any ordinal α, the ordinal α+1 (the successorMathworldPlanetmathPlanetmathPlanetmath of α) is defined to be α{α}.

  • Given a set A of ordinals, aAa is an ordinal.

If an ordinal is the successor of another ordinal, it is an successor ordinal. If an ordinal is neither 0 nor a successor ordinal then it is a limit ordinal. The first limit ordinal is named ω.

The class of ordinals is denoted 𝐎𝐧.

The von Neumann ordinals have the convenient property that if a<b then ab and ab.

Title von Neumann ordinal
Canonical name VonNeumannOrdinal
Date of creation 2013-03-22 12:32:37
Last modified on 2013-03-22 12:32:37
Owner Henry (455)
Last modified by Henry (455)
Numerical id 11
Author Henry (455)
Entry type Definition
Classification msc 03E10
Synonym ordinal
Related topic VonNeumannInteger
Related topic ZermeloFraenkelAxioms
Related topic OrdinalNumber
Defines successor ordinal
Defines limit ordinal
Defines successor