Initial ordinal

An initial ordinal is an ordinal not equinumerous with any smaller ordinal. It is therefore the canonical ordinal representative of a cardinal.

The alephs enumerate the infinite initial ordinals: ${\aleph }_0=\omega$, successor indices select the next initial ordinal, and limit indices select the least initial ordinal above all earlier values.

Assuming the axiom of choice, the well-ordering theorem makes every set equipotent to an initial ordinal. The infinite initial ordinals are exactly the ${\aleph }_{\alpha }$, so every infinite cardinal is ${\aleph }_{\alpha }$ for a unique ordinal $\alpha$.