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The cofinality of a linear order is the least cardinality of a cofinal subset. It is preserved by order isomorphisms. The order sum of ${\omega }_1$ copies of the rational order has cofinality ${\aleph }_1$, whereas appending one final rational block changes its cofinality to ${\aleph }_0$ without changing its cardinality or the dense-linear-order axioms.

For an infinite cardinal regarded as an ordinal, its cofinality $\theta =\mathrm{cf}(\kappa )$ is the least cardinality of a cofinal subset of $\kappa$. Equivalently there is a family $({\alpha }_i{)}_{i<\theta }$ of ordinals below $\kappa$ whose supremum is $\kappa$, and no smaller indexed family has that property. Since an infinite initial ordinal is a limit ordinal, $\theta$ is infinite and in particular nonzero.

Use the diagonal mechanism of König theorem for cardinal numbers directly. Suppose $F:\kappa \to {}^{\theta }\kappa$ were surjective. For each $i<\theta$, the sethas cardinality at most $|{\alpha }_i|<\kappa$. Choose the least $b_i\in \kappa \setminus E_i$. The resulting function $b:\theta \to \kappa$ is not in the range: for any $\alpha <\kappa$, cofinality supplies $i$ with $\alpha <{\alpha }_i$, and then $b_i\ne F(\alpha )(i)$.

Thus no such surjection exists. Constant functions provide an injection $\kappa \to {}^{\theta }\kappa$, and an injection in the reverse direction would give a surjection by the same inverse-and-default construction as in part (i). ConsequentlyThis argument also handles $\kappa ={\aleph }_0$, when cofinal initial segments are finite; it does not incorrectly assume that smaller cardinals themselves must form a cofinal sequence in every infinite cardinal.

First give the recursive definitions, keeping the right argument as the recursion variable. For ordinals $\alpha ,\beta$ and nonzero limit ordinals $\lambda$,Here $\xi +1$ is the successor ordinal, and the supremum of a set of ordinals is its union. These are ordinal arithmetic, defined by transfinite recursion, and differ from cardinal sum and ordinal multiplication.

Associativity of ordinal addition is true. Fix $\alpha ,\beta$ and use transfinite induction on $\gamma$. The case $\gamma =0$ is immediate. In the successor step the recursive definitions and the induction hypothesis giveFor a nonzero limit ordinal $\lambda$, the ordinals $\beta +\xi$ for $\xi <\lambda$ form an increasing cofinal subset of the limit ordinal $\beta +\lambda$. The limit clause for ordinal addition, restricted to this cofinal subset, givesCofinal restriction is valid because ordinal addition by a fixed ordinal on the left is increasing and continuous in its right argument, directly from its recursive definition. This completes the induction.