Even and odd ordinals

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In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite induction proofs.

The literature contains a few equivalent definitions of the parity of an ordinal α:

  • Every limit ordinal (including 0) is even. The successor of an even ordinal is odd, and vice versa.[1][2]
  • Let α = λ + n, where λ is a limit ordinal and n is a natural number. The parity of α is the parity of n.[3]
  • Let n be the finite term of the Cantor normal form of α. The parity of α is the parity of n.[4]
  • Let α = ωβ + n, where n is a natural number. The parity of α is the parity of n.[5]
  • If α = 2β, then α is even. Otherwise α = 2β + 1 and α is odd.[5][6]

Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number

(ω + 3)2 = (ω + 3) + (ω + 3) = ω + (3 + ω) + 3 = ω + ω + 3 = ω2 + 3

is not even.

A simple application of ordinal parity is the idempotence law for cardinal addition (given the well-ordering theorem). Given an infinite cardinal κ, or generally any limit ordinal κ, κ is order-isomorphic to both its subset of even ordinals and its subset of odd ordinals. Hence one has the cardinal sum κ + κ = κ.[2][7]


  1. ^ Bruckner, Andrew M., Judith B. Bruckner, and Brian S. Thomson (1997). Real Analysis. p. 37. ISBN 0-13-458886-X. 
  2. ^ a b Salzmann, H., T. Grundhöfer, H. Hähl, and R. Löwen (2007). The Classical Fields: Structural Features of the Real and Rational Numbers. Cambridge University Press. p. 168. ISBN 0-521-86516-6. 
  3. ^ Foran, James (1991). Fundamentals of Real Analysis. CRC Press. p. 110. ISBN 0-8247-8453-7. 
  4. ^ Harzheim, Egbert (2005). Ordered Sets. Springer. p. 296. ISBN 0-387-24219-8. 
  5. ^ a b Kamke, Erich (1950). Theory of Sets. Courier Dover. p. 96. ISBN 0-486-60141-2. 
  6. ^ Hausdorff, Felix (1978). Set Theory. American Mathematical Society. p. 99. ISBN 0-8284-0119-5. 
  7. ^ Roitman, Judith (1990). Introduction to Modern Set Theory. Wiley-IEEE. p. 88. ISBN 0-471-63519-7.