This web page concerns a remarkable theorem on cardinal arithmetic due to the Hungarian Mathematician Gyula Kőnig, or Julius König as he was known when writing in German. (This is not the same person as Dénes Kőnig after whom König's lemma—also called König's infinity lemma—is named.) Julius König devised his theorem to tackle the continuum problem, whether , and solved the problem negatively by quoting and using a result of Berstein. Unfortunately for König, Berstein's result was found to contain an error and was not true in the generality that König required.
Suppose is a set and for each a set is given, i.e. there is a (set-)function given. The sum of the sets is . The cartesian product of the sets is .
Although these definitions are valid in the absence of Choice, the Axiom is Choice is required even to show that an arbitary cartesian product or sum of sets is nonempty. We shall assume Choice throughout this web page.
We write to mean that there is an injection from to but no injection in the other direction, that is that .
Suppose sets and are given for , and suppose for all . Then .
A special case of this is when and and . Then and so König's theorem says that which was proved earlier by a sort of diagonalisation. It is therefore no surprise that a diagonalisation is required here too.
For one direction, we must define an injection from to . By AC there is a family of injections (or more precisely some function on domain such that is such an injection for each ) and by assumption none of these is onto. So my another application of AC there is for each some . We can therefore define by where and if and otherwise. if it happens that then in which case or and whereas so implies as is injective.
For the other, more interesting, direction we must show there is no injection from to . Suppose is suchj a map: we construct some with no possible value .
We start by defining where by if there is with . The set is the set of all for which this is defined. Note that as is an injection there is at most one such if there is any at all, so is well-defined.
Of course cannot be onto, as that would mean by AC that there is an injection . Therefore, by Choice again, we may take for each some . Thus and hence for some and . But this is impossible as then contradicting the choice of . So there is no such , as required.
The remarkable thing about König's inequality is that it gives us a strict inequality between the product and the sum sets: normally all that can be expected is a less-than-or-equal-to result.
Assuming the Axiom of Choice to make the following well-defined, we have
Suppose is a set and for each cardinalities is given, then the cardinal sum of the is the cardinality of any where for each and the cardinal product of the , i.e. , is the cardinality of any such .
Then König's inequality says
whenever for all .
and it follows that for if they were equal we would have