The omega fixed point is a small uncountable ordinal. When referred to as a cardinal, it is also called the aleph fixed point. It is defined as the first fixed point of the normal function \(\alpha \mapsto \omega_\alpha\), which is defined like so: * \(\omega_0 = \omega\) * \(\omega_{\alpha + 1} = \min\{x \in ext{On} : |x| > |\omega_\alpha|\}\) (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)