König’s theorem
König’s Theorem is a theorem of cardinal arithmetic.
Theorem 1.
Let and be cardinals, for all in some index set .If for all , then
The theorem can also be stated for arbitrary sets, as follows.
Theorem 2.
Let and be sets, for all in some index set .If for all , then
Proof.
Let be a function.For each we have ,so there is some that is not equal to for any .Define by for all .For any and any ,we have , so .Therefore is not in the image of .This shows that there isno surjection from onto .As is nonempty,this also means thatthere is no injection from into .This completes 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 and for all .
Also note that Theorem 2 is equivalent (in ZF) to the Axiom of Choice
, 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.)
Title | König’s theorem |
Canonical name | KonigsTheorem |
Date of creation | 2013-03-22 14:10:21 |
Last modified on | 2013-03-22 14:10:21 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 15 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 03E10 |
Synonym | Koenig’s theorem |
Synonym | Konig’s theorem |
Synonym | König-Zermelo theorem |
Synonym | Koenig-Zermelo theorem |
Synonym | Konig-Zermelo theorem |
Related topic | CantorsTheorem |