Aleph Number

Definition 1. Alpeh numbers or alephs are defined using Transfinite Recursion:

1. \[\aleph_0 = \omega.\]

2. For all ordinals $\alpha$,

   \[\aleph_{\alpha + 1} = \aleph_\alpha^+.\]

3. For limit ordinals $\alpha$,

   \[\aleph_\alpha = \sup\{\aleph_\beta : \beta < \alpha\}.\]

Definition 2. For every ordinal $\alpha$, we also define

\[\omega_\alpha = \aleph_\alpha.\]

$\aleph_\alpha$ is used when we treat it as an aleph, $\omega_\alpha$ when we treat it as an ordinal.

Proposition 1. Any infinite set $X$ is well-orderable if and only if $\lvert X \rvert$ is an aleph.

Proposition 2. Every infinite cardinal is an aleph.