User blog comment:Bubby3/The Great Googology Todo List/@comment-35470197-20190524144920/@comment-35470197-20190524224805

> First of all, your notation seems like a varaiton of the transendial integer, but is too complicated for me to understand it.

Then I am certain that you do not understand thet system of transcendental integers, because the system of transcendental integer is as difficult as it... I recommend you not to use terminology as if you understood the notion.

> Also, the ordinal notation is impossible to use in practice

Then I am certain that you do not understand the definitions of PTO and an ordinal notation, because the notation of ordinals is not an ordinal notation by definition. I recommend you not to use terminology as if you understand the notion.

> BMS catching function is just asking for when the growth rates of the "n^2" and "constant" varations are comparable, (to some definition of comparable).

What comparability do you use? It is a very important point.

> This doesn't need to be an ordinal notation, just a notation with an associate set of fundemental sequnces, and BMS is a function which will evuantlly evaulate to numbers.

Since it is not an ordinal notation, BMS does not necessarily eventually evaluate to numbers. It can cause loops unlike an ordinal notation, although it is believed to be well-founded without any reasons. Therefore you need to verify the termination first.

So "the great googology to do list" for you is the following: Just work in a step-by-step way. Skipping is not "the great googology". Reagrding stuffs which you could not understand to be meaningless is not, either.
 * 1) Understand the transcendental integers in a precise (not intuition-based) way.
 * 2) Understand the definitions of PTO and an ordinal notation in a precise (not intuition-based) way.
 * 3) Formally analyse pair sequences, and after then, BMS.
 * 4) Define the comparability so that your "BMS catching function" is actually total.