User blog comment:Simplicityaboveall/Extremely Large Numbers 4/@comment-5529393-20160801195700

I'm afraid I have to repeat the common refrain: Despite all that you've written here and in Extremely Large Numbers 3, you still haven't defined what it means to "write m in complete hereditary form in base n". Until you do that, everything you write about how far your notation extends is just speculation and intuition.

You write "we have to do some approximations to simplify some expressions." You can do that, _after_ you have defined your terminology, otherwise your expressions don't actually mean anything and can't be approximated.

Also, you can't justify (w^^w)^^w ~ w^^(w2) by making an analogy with finite numbers. You can make a lot of mistakes that way - for instance, one might suppose that since 2^n < aleph_0 for all finite n, then 2^aleph_0 must equal aleph_0, but that's not true. Anyway, the first step is to define what ^^ means for transfinite ordinals.

Good luck on further refining your notation.