Omega fixed point

The omega fixed point is a small uncountable ordinal. When referred to as a cardinal, it is also called the aleph fixed point. To define the omega fixed point, we first define the following hierarchy:


 * \(\omega_0 = \omega\)
 * \(\omega_{\alpha + 1} = \min\{x \in \text{On} : |x| > |\omega_\alpha|\}\) (the smallest ordinal with cardinality greater than \(\omega_\alpha\))
 * \(\omega_\alpha = \sup\{\beta < \alpha : \omega_\beta\}\) for limit ordinals \(\alpha\) (the limit of all smaller members in the hierarchy)

Then the omega fixed point is the first fixed point of the function \(\alpha \mapsto \omega_\alpha\), visualized as \(\omega_{\omega_{\omega_{._{._.}}}}\).