User blog comment:P進大好きbot/Ordinal Notation Associated to a Proper Class of Ordinals/@comment-25601061-20180625142705/@comment-35470197-20180625145524

1. I just used limit points of a class of ordinal numbers. For a class \(Y \subset \textrm{ON}\), an \(\alpha \in \textrm{ON}\) is said to be a limit point of \(Y\) if \(\alpha = \bigcup \{\alpha' \in \alpha \mid \alpha' \in Y\}\) holds.

2. In general, for a \(y_0 \in \textrm{ON}\) and a strictly increasing endo-function \(F(x)\) of \(\textrm{ON}\) (given as a function formula \(\Phi(x,y)\) in ZFC set theory whose domain contains \(\textrm{ON}\), i.e. \(\textrm{ZFC} \vdash \forall x \in \textrm{ON}, \exists ! y, \Phi(x,y)\)), there is a function \(g\) satisfying the following: Moreover, \(g\) is unique, i.e. the function formula \(\Phi'\) presenting \(g\) is unique up to provable equivalence.
 * \(\textrm{ZFC} \vdash g(0) = y_0\)
 * \(\textrm{ZFC} \vdash \forall x \in \textrm{ON}, g(x \cup \{x\},z) = f(g(x))\)
 * \(\textrm{ZFC} \vdash \forall x \in \textrm{ON} \setminus \{0\}, (x = \bigcup_{x' \in x} x') \to g(x) = \bigcup_{x' \in x} g(x')\)

I note that this statement is a special consequence of the original transcendental induction in ZFC set theory. I mean, there is a wider formulation of the transcendental induction. The transcendental induction with two variables also works in the same way.