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.
See also[]
Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function) · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)
References[]
- ↑ M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic, Volume 29, Issue 4, pp. 249--263, 1990.