Aleph Number

DEF-ALF. Aleph Number.

• We define the alpeh numbers / alephs using Transfinite Recursion:

  1. \[\aleph_0 = \omega;\]
  2. \[\aleph_{\alpha+1} = \aleph_\alpha^+;\]
  3. \[\aleph_\alpha=\sup\{\aleph_\beta:\beta<\alpha\}\]

     for every limit ordinal $\alpha$.

• We also write $\omega_\alpha=\aleph_\alpha$. $\aleph_\alpha$ is used when referring to an aleph number, $\omega_\alpha$ when referring to an ordinal.

PROP-ALF-WO.

For any infinite set $X$, $X$ is well-orderable if and only if $\lvert X\rvert$ is an aleph.

PROP-ALF-CA. $\lrimp\AC$

Every infinite cardinal is an aleph.