Omega fixed point

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.M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic, Volume 29, Issue 4, pp. 249--263, 1990. 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.
== See also ==
ja:オメガ収束点
== References ==
Category:Transfinite ordinals
Category:Set theory