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单词 KonigsTheorem
释义

König’s theorem


König’s Theorem is a theorem of cardinal arithmetic.

Theorem 1.

Let κi and λi be cardinals, for all i in some index setMathworldPlanetmathPlanetmath I.If κi<λi for all iI, then

iIκi<iIλi.

The theorem can also be stated for arbitrary sets, as follows.

Theorem 2.

Let Ai and Bi be sets, for all i in some index set I.If |Ai|<|Bi| for all iI, then

|iIAi|<|iIBi|.
Proof.

Let φ:iIAiiIBi be a function.For each iI we have |φ(Ai)||Ai|<|Bi|,so there is some xiBithat is not equal to (φ(a))(i) for any aAi.Define f:IiIBiby f(i)=xi for all iI.For any iI and any aAi,we have f(i)(φ(a))(i), so fφ(a).Therefore f is not in the image of φ.This shows that there isno surjection from iIAi onto iIBi.As iIBi is nonempty,this also means thatthere is no injection from iIBi into iIAi.This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof of Theorem 2.Theorem 1 follows as an immediate corollary.∎

Note that the above proof is a diagonal argument,similar to the proof of Cantor’s Theorem.In fact, Cantor’s Theorem can be considered as a special case of König’s Theorem, taking κi=1 and λi=2 for all i.

Also note that Theorem 2 is equivalentMathworldPlanetmathPlanetmathPlanetmath (in ZF) to the Axiom of ChoiceMathworldPlanetmath, as it implies that products (http://planetmath.org/GeneralizedCartesianProduct) of nonempty sets are nonempty. (Theorem 1, on the other hand, is not meaningful without the Axiom of Choice.)

TitleKönig’s theorem
Canonical nameKonigsTheorem
Date of creation2013-03-22 14:10:21
Last modified on2013-03-22 14:10:21
Owneryark (2760)
Last modified byyark (2760)
Numerical id15
Authoryark (2760)
Entry typeTheorem
Classificationmsc 03E10
SynonymKoenig’s theorem
SynonymKonig’s theorem
SynonymKönig-Zermelo theorem
SynonymKoenig-Zermelo theorem
SynonymKonig-Zermelo theorem
Related topicCantorsTheorem
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