Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 5 / i / Solution

König's theorem for cardinal numbers states that, for a set-indexed family with ${\kappa }_i<{\lambda }_i$ at every index,Here the sum is the cardinality of a disjoint union and the product is the cardinality of a Cartesian product. Work in ZFC. Choose disjoint sets $A_i$ of sizes ${\kappa }_i$ and sets $B_i$ of sizes ${\lambda }_i$. For any function $f:{⨆}_i{A}_i\to {\prod }_i{B}_i$, its $i$th coordinate values on $A_i$ form a subset of $B_i$ of cardinality at most ${\kappa }_i$. Choose $b_i\in B_i$ outside that subset, using choice. Then $b=(b_i{)}_i$ cannot equal $f(a)$ for any $a$: if $a\in A_j$, its $j$th coordinate was specifically excluded. Thus there is no surjection from the disjoint union onto the product.

Cardinal comparability under choice now implies the displayed strict inequality. Empty index set gives $0<1$ directly; empty individual $A_i$ cause no difficulty, since the corresponding coordinate excludes an empty image. This is the cardinal diagonal theorem, rather than the finitely branching tree result also bearing König's name.