Aleph-n

Definition

Let $n$ be an ordinal. Then ${\aleph }_n$ is defined inductively as follows:

$$\begin{array}{r}{\aleph }_0=\omega \\ {\aleph }_{\alpha +}=({\aleph }_{\alpha }{)}^{+}\\ {\aleph }_{\gamma }=\bigcup \{{\aleph }_{\alpha }.\alpha <\gamma \}\text{ for γ limit ordinal}\end{array}$$

Where $A^{+}$ is the next largest well-ordered cardinal.

As such all ordinals ${\aleph }_{\alpha }$ are canonically well-ordered, and therefore equated with some ordinal.