User blog comment:Flitri/An ordinal Collapsing up to the Least weakly Mahlo Cardinal/@comment-35470197-20190409053305/@comment-35470197-20190410220613

> cof(x) = min{μ|α[μ] = x for some μ-indexed sequence α with α[δ] < α[λ] ⇌ δ < λ}

It is incorrect, because any x admits such a singleton seuqence (x). You need to write cof(x) = min {μ | sup_{β<μ} α[β] = x for some μ-indexed sequence α with α[δ] < α[λ] ⇌ δ < λ}

> N = Enum{x | x < ω}

Uh, you are confounding {x | P(x)} and Enum{x | P(x)}. You should remobe "Enum" from the definitions of the four stuffs, i.e. N, Reg, Lim, and Suc. They are not enumeration functions, but specified (in the sense of comprehension schema) subclasses of Ord.

The descriptions of Enum below the four definitions are good.

> If we have two sign sequences A & B as well as two ordinals X & Y:

I could not understand any relation between A & B and X & Y. Also, what are Xc & Yc? Also, you have never defined the notion of a sign sequence in a recursive way. If you allow any sequence of signs, the conditions do not work because < relation on sequences is not a total order.

> α^^X = α^(α^^(X-1)) & α^^0 = 1

What is X-1?

Well, as I have pointed out so many undefined stuffs, you are using many non-traditional conventions without specifying all precise definitions. It is very hard to point out "all" problems. I hope that you will fix all such problems. After then, I can point out more (non-syntax theoretic but set theoretic) problems.

The most important point is that you have never written any theoretic reason why you go beyond weakly Mahlo. Using weakly Mahlo itself does not ensure its strength, and hence you need to show the reason. Since Rathjen's mathod is very smart, it is not obvious for others why you can skip such techniques.

Or you just state that you have defined a function collapsing M, but not that your collapsing function is as strong as standard OCFs with weakly Mahlo, right?

Also, your notation is not an ordinal notation, because you have notexplicitly defined the recursive well-ordering on the notation. Please read a detailed reason written in my blog post. Many of the problems in your definition are written here, and hence it is better for you to read it before asking other problems.