On

\(\textrm{On}\) (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 .
* 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 \(\textrm{On}\).K. Kunen, Set Theory: An Introduction to Independence Proofs, Studies in Logic and the Foundations of Mathematics, Volume 102, North Holland, 1983. 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 \(\omega\).
== References ==
== See also ==
Category:Set theory
Category:Proper class
Category:Transfinite ordinals