Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2008 / B / 7 / i / Solution

König theorem for cardinal numbers states that if ${\kappa }_i<{\lambda }_i$ for each index $i\in I$, thenRepresent each cardinal by its initial ordinal, let $A=\{(i,\alpha ):i\in I,\alpha <{\kappa }_i\}$, and let $B={\prod }_{i\in I}{\lambda }_i$.

First there is an injection $A\to B$: encode $(i,\alpha )$ by the function which is zero away from $i$ and takes value $\alpha +1$ at $i$. The nonzero coordinate identifies $i$, and its value identifies $\alpha$. It is valid because $\alpha +1\le {\kappa }_i<{\lambda }_i$; an index with ${\kappa }_i=0$ contributes no domain points.

For strictness suppose a surjection $F:A\to B$ existed. At coordinate $i$, the values $F(i,\alpha )(i)$ for $\alpha <{\kappa }_i$ occupy fewer than ${\lambda }_i$ possibilities. Choose $b_i$ as the least ordinal below ${\lambda }_i$ absent from that set. The resulting $b=(b_i)\in B$ differs from $F(i,\alpha )$ at coordinate $i$ for every $(i,\alpha )\in A$, contradicting surjectivity.

An injection $B\to A$ would produce a surjection $A\to B$ by inverting on its image and assigning the zero function elsewhere. It is therefore impossible. The injection in the first direction and impossibility of the reverse give the strict cardinal inequality. This diagonal argument uses the canonical well-orders of the specified cardinals to choose missing coordinate values, rather than an unexplained arbitrary choice at that step.