Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2007 / B / 7 / i / Solution

A cardinal number is an initial ordinal: an ordinal not in bijection with any smaller ordinal. In $ZFC$ every set has the cardinality of a unique such ordinal.

For an infinite cardinal $\kappa$, its cofinality is the least cardinality of an unbounded subset of the ordinal $\kappa$. Equivalently,A subset is unbounded when no ordinal below $\kappa$ bounds all its elements. A cofinal sequence can be taken increasing, and $\mathrm{cf}(\kappa )$ is an infinite regular cardinal.