In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set,
The sum here is the cardinality of the disjoint union of the sets mi, and the product is the cardinality of the Cartesian product.
However, without the use of the axiom of choice, the sum and the product cannot be defined as cardinal numbers, and the meaning of the inequality sign would need to be clarified.
Kőnig's theorem was introduced by Kőnig (1904) in the slightly weaker form that the sum of a strictly increasing sequence of nonzero cardinal numbers is less than their product.
where < means strictly less than in cardinality, i.e. there is an injective function from Ai to Bi, but not one going the other way.
The union involved need not be disjoint (a non-disjoint union can't be any bigger than the disjoint version, also assuming the axiom of choice).
In this formulation, Kőnig's theorem is equivalent to the axiom of choice.
[1] (Of course, Kőnig's theorem is trivial if the cardinal numbers mi and ni are finite and the index set I is finite.
Kőnig's theorem is remarkable because of the strict inequality in the conclusion.
There are many easy rules for the arithmetic of infinite sums and products of cardinals in which one can only conclude a weak inequality ≤, for example: if
, where the index set I is the natural numbers, yields the sum
, then the left side of the above inequality is just
(Historically of course Cantor's theorem was proved much earlier.)
One way of stating the axiom of choice is "an arbitrary Cartesian product of non-empty sets is non-empty".
Let Bi be a non-empty set for each i in I.
Thus by Kőnig's theorem, we have: That is, the Cartesian product of the given non-empty sets Bi has a larger cardinality than the sum of empty sets.
Thus it is non-empty, which is just what the axiom of choice states.
Since the axiom of choice follows from Kőnig's theorem, we will use the axiom of choice freely and implicitly when discussing consequences of the theorem.
Kőnig's theorem has also important consequences for cofinality of cardinal numbers.
If κ is regular, then this follows from Cantor's theorem.
Choose a strictly increasing cf(κ)-sequence of cardinals approaching κ.
We finish the proof by showing that λ = κ.
According to Easton's theorem, the next consequence of Kőnig's theorem is the only nontrivial constraint on the continuum function for regular cardinals.
Suppose that, contrary to this corollary,
κ ≥ cf ( μ )
Assuming Zermelo–Fraenkel set theory, including especially the axiom of choice, we can prove the theorem.
The axiom of choice implies that the condition A < B is equivalent to the condition that there is no function from A onto B and B is nonempty.
So we are given that there is no function from Ai onto Bi≠{}, and we have to show that any function f from the disjoint union of the As to the product of the Bs is not surjective and that the product is nonempty.
That the product is nonempty follows immediately from the axiom of choice and the fact that the factors are nonempty.
Then the product of the elements bi is not in the image of f, so f does not map the disjoint union of the As onto the product of the Bs.