Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 16H / b / Solution

Apply the stated cardinal comparability principle to $x$ and its Hartogs ordinal $h(x)$. The Hartogs theorem rules out an injection $h(x)\to x$, so comparability supplies an injectionOrder $x$ by $u\prec v$ exactly when $j(u)\in j(v)$. This is the restriction of an ordinal well-order and therefore well-orders $x$. Hence every set can be well-ordered. The well-ordering theorem is equivalent to the axiom of choice: for a family of nonempty sets, well-order its union and choose the least member of each set. The comparison statement therefore implies choice.