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

Since I am also working with my OCF and my associated ordinal notation system with weakly Mahlo cardinal, I am interested in your work. But your description looks a little ambiguous. Could you answer the following questions and comments?

> Enum{ x | P(x,α)} is the set of all ordinals satisfying P enumerated by α.

What does "the set ... enumerated by α" mean? You mean that Enum{ x | P(x,α)} is just a function Ord -> Ord Enumerating {x ∈ Ord | P(x)}, right? Or do you have an occurrence of a parameter α in P?

> Enum[min]{ x | P(x,α)} is the set of all ordinals satisfying P enumerated by α such that their distance is the smallest.

What does the distance mean? Recall that Odr is not metrisable. Is it just the substraction of ordinals?

> Formally: ψ(α) = min{ x | x not in K(α) over F}

It is informal. What does "x not in K(α) over F" mean? Also, F is not a set of functions, because you defined it as the images of functions.

> L(ν) is the associated limit ordinal for the successor ordinal ν: L(N) → ω & L(ω+N) → ω*2 etc.

I could not understand what you meant. N is not a successor ordinal. Could you write down the precise definition of L?

> Ω(α+1) = min{x ||x| = Ω(α+1)}—|X| is the associated cardinal of the ordinal X

Circular logic. It does not work.

> Ω(ν-1) <  ψ(ν, α)  <  Ω(ν)  <  Ω(L[ν]) & cof(ν) = 1

What does L[ν] mean? Is it just L(ν)?

> K[0](α, β, γ) = {α, β, γ} ⋃ {Ω[1]}

What does Ω[1] mean? Is it just Ω(1)?

> Δ[Φ,St](x) is any function Δ: Ord → Ord such that Δ(Φ,St](x) = α → Φ(St,α) = 1 for the statement St

You are not allowed to use truth predicate of a statement in first order set theory. It does not work.

> Rules when p = 0 and F[θ](x) is the enumerated function based on η:

They do not work because the expressions in the left hand sides are not unique. Please recall why Rathjen introduced normal forms with respect to expressions by specific functions. So it is ill-defined.