User blog comment:Bubby3/The Great Googology Todo List/@comment-35470197-20190524144920/@comment-24725252-20190524152931

First of all, your notation seems like a varaiton of the transendial integer, but is too complicated for me to understand it, at least right now. Also, the ordinal notation is impossible to use in practice, because finding proofs is extremely difficult, and that even smallish ordinals will turn out to be extremely difficult to express.

For  the formal proof of pair sequence system, I am going to do that, and make a new ntation that is just a restatment of pair sequnces and prove that it is, and prove correspondence against another ntatoin that no one questioned wheater it terminates before, or to bucholoz hydra

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). 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. The "ordinal notation" associated is just a varation of BMS. If BMS is well-founded, then we can compare the growh rates of both varations of them, probally on BM2.3.