User blog comment:Flitri/An ordinal Collapsing up to the Least weakly Mahlo Cardinal/@comment-35470197-20190409053305/@comment-35470197-20190410054653

Well, your definition does not work for your purpose. For example, consider the equality ψ[S](ν, α) = Enum{ x | x in Lim} < Ω(ν). Then θ(-) would be ψ[S](ν,-), and P wpuld be "x \in Lim" without the occurence of β. Then the resulting function is just the enumeration function of limit ordinals, and hence you do not have the upper bounde Ω(ν).

I guess that what you intended as the definition of \(\theta\) for a formulae \(P\) is \(\theta(\alpha\) = \min \{ x | \forall \beta < \alpha, \theta(\beta) < x \land P(x,\theta(\beta))\}\) with respect to the traditional convention. It is not a usual enumeration function, and hence is not described as Enum. On the other hand, if P is a formula without the occurence of β, then θ coincdes with the enumeration function Enum{x | P(x)}. Therefore I guess that you confound these two distinct notions.

For example, how about writing definitions in the following way? You failed because you simultaneously defined these two functions in a single convention.
 * 1) Enum{x | P(x)} is the enumeration function of the class {x | P(x)}, where P is a formula with a unique free variable x,
 * 2) and Enum{x | P(x,β)} is the function \(\theta\) given by setting \(\theta(\alpha) := \min \{x | \forall \beta < \alpha, \theta(\beta) < x \land P(x,\theta(\beta))) in the recursive way, where O is a formula with occurrence of free variables x and β.