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The omega fixed point is an ordinal which is the fixed point of the normal function $$\alpha \mapsto \omega_\alpha$$, or specifically the smallest such ordinal. When referred to as a cardinal, it is also called the aleph fixed point. $$\omega_\alpha$$ is defined as:

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

The enumeration function of the class of omega fixed points is denoted by $$\Phi_1$$ using Rathjen's Φ function.[1] In particular, the least omega fixed point can be expressed as $$\Phi_1(0)$$.

The omega fixed point is most relevant to googology through ordinal collapsing functions. It is often expressed as $$\psi_I(0)$$ using an unspecified or undefined $$\psi$$ function, and the first inaccessible cardinal $$I$$. For the issue on the $$\psi$$ function, see the main article on countable limit of Extended Buchholz's function.