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

An initial ordinal is an ordinal $\kappa$ that is not equipotent to any smaller ordinal. Equivalently, it is the least ordinal having its cardinality. The aleph numbers enumerate the infinite initial ordinals:${\aleph }_{\alpha +1}$ is the least initial ordinal greater than ${\aleph }_{\alpha }$, and at a limit $\lambda$, ${\aleph }_{\lambda }$ is the least initial ordinal greater than every ${\aleph }_{\beta }$ for $\beta <\lambda$.

Each ${\aleph }_{\alpha }$ is infinite and initial by construction. Conversely, let $\kappa$ be an infinite initial ordinal. The infinite initial ordinals below $\kappa$ form a set and therefore have an ordinal order type $\alpha$. The recursive enumeration above lists exactly those predecessors before stage $\alpha$, so its next value is ${\aleph }_{\alpha }=\kappa$.

Read literally, “has cardinality ${\aleph }_{\alpha }$” is too weak: $\omega +1$ has cardinality ${\aleph }_0$ but is not initial. The valid characterization isIf “has cardinality” in the question means “is the cardinal represented by,” this is precisely the requested statement.