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:

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]

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References

  1. ↑ Bruckner, Andrew M.; Judith B. Bruckner & Brian S. Thomson (1997). Real Analysis . pp.  37. ISBN   0-13-458886-X.
  2. 1 2 Salzmann, H., T. Grundhöfer, H. Hähl, and R. Löwen (2007). The Classical Fields: Structural Features of the Real and Rational Numbers. Cambridge University Press. pp.  168. ISBN   0-521-86516-6.CS1 maint: multiple names: authors list (link)
  3. ↑ Foran, James (1991). Fundamentals of Real Analysis. CRC Press. pp.  110. ISBN   0-8247-8453-7.
  4. ↑ Harzheim, Egbert (2005). Ordered Sets . Springer. pp.  296. ISBN   0-387-24219-8.
  5. 1 2 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. pp.  88. ISBN   0-471-63519-7.
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