User blog comment:Mh314159/Natural number recursion - first 4 rule sets/@comment-35470197-20191019143654/@comment-35470197-20191024225717

> I have a hugely busy life and don't have the time

I see. But I guess that studying elementary intuition of ordinals of the current level can be done within one hour since you are now understanding how to add \(\omega\). Instead of extending functions three times, you can perhaps learn it. (Of course, it is rasonable for everyone to respect his or her preference, e.g. to create a notation without an idea from ordinals.)

> Let's ignore the alpha superscripts which I thought was adding a lot because it recursed the argument so many times to put the entire previous structure into the subscript.

One of the most important fact is that "to recurse in a single pattern" such as to put the entire previous structure so many times into indices just causes a single effect to an ordinal, which might be \(+2\), \(+\omega+1\), or \(+\omega^n\) up to now. Therefore you need a new type of a stronger recursion.

> The next plan would be to define bracket strings that recurse the entire structure back into the brackets.

I am looking forward to seeing it. As I noted, it is good to create a new blog post, because rules separated into many disjoint comments are difficult to read. (Also, comments are not visible from several environments such as my mobile phone.)