User blog comment:Flitri/An ordinal Collapsing up to the Least weakly Mahlo Cardinal/@comment-35470197-20190409053305/@comment-35470197-20190409222858

> I have now changed a few things, along with a new link.

Ok. But if you need sufficient feedbacks, how about copying manuscript in the body of your blog post? It is really hard for others to recognise all changes of articles in non-parmanent external links. In this wiki, we can refer to the history of editting, and hence can easily grasp the full changes.

> 1) [Ordinal Distance] Distance really just means the next smallest, I think I just got confused as to what term to use.

Then could you write down the precise definitions of Enum{ x | P(x,α)} and Enum[min]{ x | P(x,α)}? Well, this comment would not look to be questions, I am asking this third time. In order to define a stuff in set theory, all conventions but traditional ones should be explicitly clarified in order to avoid intuition-based arguments.

> 2) [Fundamental Sequences] I am confused when you say the rules for Fundamental sequences are ill-defines could you provide specific examples? I’ll get around to canonical sequences as well.

For example, a system of fundamental sequences below \(\varepsilon\) expressed in the following way is ill-defined in the same reason: You need to restrict all the expressions to Cantor normal forms.
 * 1) \((\alpha + \beta)[n] = \alpha + (\beta[n])\) (\(beta\) is a non-zero limit ordinal)
 * 2) \((\alpha \times \beta)[n] = \alpha \times (\beta[n])\) (\(\beta\) is a non-zero limit ordinal)
 * 3) \(\omega[n] = n\)
 * 4) \(\omega^{\alpha + 1}[n] = \omega^{\alpha} \times n\)
 * 5) \(\omega^{\alpha}[n] = \omega^{\alpha[n]}\) \(\(\alpha\) is a non-zero limit ordinal)

For a more general situation, you need to define normal forms. If you do not know how to do so, please read the papers of Buchholz or Rathjen. The method is clearly described. People tend to think that it might be difficult for them before they actually read them, but I am certain that at least you can easily read it because you have studied highly complicated set theoretic stuffs such as weakly Mahlo cardinal.