User:Emlightened/not-so-secret future blog posts

Eh; read the title. Most of this is just for editing remotely before copying into a blog post, instead of having to wait to get back to my laptop.

Why isn't this a thing?
Seriously, I want to know why nobody uses this diagonal.

To generalize polynomials, we make arrays or numbers.

To generalize arrays of numbers, we turn them into an ordinal polynomial of \(\omega\), and make the Veblen hierarchy.

To generalize the Veblen hierarchy, we turn them into an ordinal polynomial of \(\Omega\), and make ordinal collapsing functions (which we then extend across the inaccessible hierarchy).

To generalize ordinal collapsing function ordinals (specifically, the inaccessible hierarchy), we turn them into an ordinal polynomial of \(M\), and make another layer of ordinal collapsing functions (which we then extend across the Mahlo hierarchy).

To generalize ordinal collapsing function ordinals (specifically, the Mahlo hierarchy), we turn them into an ordinal polynomial of \(K\), and make another layer of ordinal collapsing functions. (See deedlit's blog.)

The only notation I have ever seen that capitalizes across this diagonal, aside from my own WIP, is possibly the Basichu matrix system (it's unclear how this diagonalises). I have also seen a user page do this to ordinals informally once, but no notations were really made.

This is obviously the next level of top-end recursively-constructed googology. It's not very new, it doesn't require someone great at ordinals to see it, so why are there no notations that use this?

As a footnote, is this the PTO of Z 2<\sub>? (the PTOs Deedlit mentions seem like they may converge to this, but it may just be a major subsystem's PTO.