## FANDOM

9,760 Pages

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:

• $$\omega_0 = \omega$$
• $$\omega_{\alpha + 1} = \min\{x \in \text{On} : |x| > |\omega_\alpha|\}$$ (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)

The omega fixed point is most relevant to googology through ordinal collapsing functions. It can itself be expressed as $$\psi_I(0)$$, using the Buchholz psi function and the first inaccessible cardinal.