idempotency of infinite cardinals

In this entry, we show that every infiniteMathworldPlanetmath cardinal is idempotentPlanetmathPlanetmath with respect to cardinal additionMathworldPlanetmath and cardinal multiplication.

Theorem 1.

κκ=κ for any infinite cardinal κ.


For any non-zero cardinal λ, we have λ=1λλλ. So given an infinite cardinal κ, either κ=κκ or κ<κκ. Let 𝒞 be the class of infinite cardinals that fail to be idempotent (with respect to ). Suppose 𝒞. We shall derive a contradictionMathworldPlanetmathPlanetmath. Since 𝒞 consists entirely of ordinalsMathworldPlanetmathPlanetmath, it is therefore well-ordered, and has a least member κ.

Let K=κ×κ. As K is a collectionMathworldPlanetmath of ordered pairs of ordinals, it has the canonical well-ordering inherited from the canonical ordering on On×On. Let α be the ordinal isomorphicPlanetmathPlanetmathPlanetmath to K. Since κ<κκ=|K|, there is an initial segment L of K that is order isomorphic to κ.

Since L is an initial segment of K, L={(β1,β2)(β1,β2)(α1,α2)} for some (α1,α2)K. The well-order denotes the canonical ordering on K. Let λ=max(α1,α2). Since LK=κ×κ, α1<κ and α2<κ, and therefore λ<κ.

For any (β1,β2)L, we have (β1,β2)(α1,α2), which implies that max(β1,β2)λ. Therefore Lλ+×λ+, or |L||λ+×λ+||λ+||λ+|. There are two cases to discuss:

  1. 1.

    If λ is finite, so is λ+×λ+, contradicting that L is (order) isomorphic to κ, an infinite set.

  2. 2.

    If λ is infinite, so is |λ+|. Since λ<κ, and κ is a limit ordinalMathworldPlanetmath, |λ+|<k as well, which means |λ+|𝒞, or |λ+||λ+|=|λ+|. Therefore |L||λ+||λ+|=|λ+|λ+<κ, again contradicting that L is (order) isomorphic to κ.

Therefore, the assumptionPlanetmathPlanetmath 𝒞 is false, and the proof is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. ∎

Corollary 1.

If 0<λκ and κ is infinite, then λκ=κ.


κ=1κλκκκ=κ. By Schroder-Bernstein’s Theorem, λκ=κ. ∎

Corollary 2.

If λκ and κ is infinite, then λ+κ=κ.


κ=0+κλ+κκ+κ=2κκκ=κ by the corollary above (since 2κ). Another application of Schroder-Bernstein gives κ=λ+κ. ∎

Since κκ, we get the following:

Corollary 3.

κ+κ=κ for any infinite cardinal.

Remark. No cardinal greater than 1 is idempotent with respect to cardinal exponentiation. This is a direct consequence of Cantor’s theorem: κ<2κκκ.

Title idempotency of infinite cardinals
Canonical name IdempotencyOfInfiniteCardinals
Date of creation 2013-03-22 18:53:30
Last modified on 2013-03-22 18:53:30
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 03E10
Related topic CanonicalWellOrdering