König's theorem (set theory)

The set of King is a set of set theory, which was discovered by the Hungarian mathematician Julius King in 1905. The set is a strict inequality between two cardinal numbers.

Statement

For a family of cardinal numbers the sum of cardinal numbers is the cardinality of the disjoint union of sets of cardinality,

And the product is the cardinality of the Cartesian product,

Here are some of the pairwise disjoint sets, for example. The well- definedness of both operations follows from the axiom of choice.

The König's theorem now states:

For two cardinal consequences and are all:

Evidence

Be, two families of pairwise disjoint sets with. Without loss of generality, we can assume that. It is to show that there is an injective but not bijective mapping

For each is an item. Be. Then there exists a unique with. Let the function

Then is injective.

Let now be given any such figure. For defining than one element. Then out at the site different from all the images. Since this applies to all, is not surjective and thus not bijective.

Conclusions

From the set of King, further inequalities can be directly derived ( and are cardinal numbers ):

• Identifies the cofinality of, then for infinite.
• For infinity.