Alpeh numbers / alephs are defined using Transfinite Recursion:
- \[\aleph_0 = \omega.\]
For all ordinals $\alpha$,
\[\aleph_{\alpha+1} = \aleph_\alpha^+.\]For limit ordinals $\alpha$,
\[\aleph_\alpha = \sup\{\aleph_\beta:\beta<\alpha\}.\]We also write $\omega_\alpha=\aleph_\alpha$. $\aleph_\alpha$ is used when referring to an aleph number, $\omega_\alpha$ when referring to an ordinal.
For any infinite set $X$, $X$ is well-orderable if and only if $\lvert X\rvert$ is an aleph.
ALF#PROP-CA. $\lrimp\AC$
Every infinite cardinal is an aleph.