Alpeh numbers or alephs are defined using Transfinite Recursion:
- \[\aleph_0 = \omega.\]
For all ordinals $\alpha$,
\[\aleph_{\alpha + 1} = \aleph_\alpha^+.\]For limit ordinals $\alpha$,
\[\aleph_\alpha = \sup\{\aleph_\beta : \beta < \alpha\}.\]For every ordinal $\alpha$, we also write
\[\omega_\alpha = \aleph_\alpha.\]$\aleph_\alpha$ is used with emphasis on an aleph number, $\omega_\alpha$ with emphasis on an ordinal.
For any infinite set $X$, $X$ is well-orderable if and only if $\lvert X \rvert$ is an aleph.
The following proposition is equivalent to $\AC$:
Every infinite cardinal is an aleph.