On

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\(\textrm{On}\) (also commonly denoted \(\textrm{Ord}\)) denotes the class of all ordinals.

Properties

• The class \(\textrm{On}\) is a proper class, i.e. is not a set, by Burali-Forti paradox.
• The class \(\textrm{On}\) is transitive, i.e. for any \(x \in \textrm{On}\), \(x \subset \textrm{On}\).
• The class \(\textrm{On}\) is well-ordered with respect to \(\in\), i.e. the restriction of \(\in\) on \(\textrm{On}\) is a strict total ordering and every non-empty subclass of \(\textrm{On}\) admits the least element with respect to \(\in\).
• The equality \(\textrm{On} = \omega\), i.e. the statement "every ordinal is finite", is independent of \(\textrm{ZF}\) set theory minus the axiom of infinity as long as it's consistent, while it contradicts \(\textrm{ZF}\) set theory.
• The equality \(\textrm{On} = \omega+\omega\), i.e. the statement "every ordinal is finite or the sum of \(\omega\) and a finite ordinal", is independent of \(\textrm{ZF}\) set theory minus the axiom of replacement as long as it's consistent, while it contradicts \(\textrm{ZF}\) set theory.

Application

By the well-foundedness of \(\textrm{On}\) with respect to \(\in\), transfinite induction is extended to \(\textrm{On}\). Therefore we do not have to find a sufficiently large ordinal in order to verify a statement on an ordinal by transfinite induction. We have many examples of maps in googology defined by transfinite induction on \(\textrm{On}\):

• The addition, the multiplication, and the power of ordinals
• Veblen function
• Ordinal collapsing functions
• The hierarchies in von Neumann universe \(V\) and constructible universe \(L\)

Kleene's fixed point theorem

It is well-known that Kleene's fixed point theorem extends to \(\textrm{On}\). For example, let \(f \colon \textrm{On} \to \textrm{On}\) be a normal function. Take an arbitrary \(\alpha \in \textrm{On}\). By the axiom of replacement, the class \(\{f^n(\alpha)\mid n \in \mathbb{N}\}\) is a set, and hence it admits the supremum \(\beta\). By the normality of \(f\), \(\beta\) is a fixed point of \(f\). This fact is used to show the totality of the Veblen function.

Beginners should be very careful about the condition of normality, because googologists often unreasonably drop it. For example, the enumeration \(\alpha \mapsto I_{\alpha}\) of weakly inaccessible cardinals under the assumption of the totality on \(\textrm{On}\) is not normal, and its iteration \(I_{I_{\cdot_{\cdot_{\cdot_{I_0}}}}}\) does not give its fixed point, because it is of cofinality \(\omega\).

See also

Ordinals, ordinal analysis, and set theory

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)