User blog comment:Simplicityaboveall/The Construction of Extremely Large Numbers/@comment-5529393-20160724103034

In order for h(m,n) to be well-defined, $$R^\omega_n(m)$$ must be well-defined, which means you must describe precisely what "complete hereditary form of the number m to base n" is. Usually "hereditary base-n notation" means to write m as a sum of powers of n, then write the exponents as sum of powers of n, and so on. Changing n to $$\omega$$ will only get you ordinals below $$\varepsilon_0$$. Clearly this is not what you mean, since you later start talking about changing $$10 \uparrow^k 10$$ to $$\omega \uparrow^k \omega$$, and then later more complicated structures. So, what exactly is the complete hereditary form of the number m to base n?

Note that Knuth arrows and transfinite ordinals don't play well under the obvious definition; using that definition, we get $$\omega \uparrow^\beta \gamma = \varepsilon_0$$ for all $$\beta \ge 3, \gamma \ge 2$$, which is not what you want. So if you use large number notations for your complete hereditary form, you need to specify how they operate on transfinite ordinals as well.