User blog comment:Tetramur/Pentational arrays and beyond - comparisons/@comment-37993808-20200108154849/@comment-35470197-20200109234712

I see. Then X^α corresponds to the "background" given as the o(X^α)-dimensional space, i.e. N^{o(X^α)}, where o(X^α) is defined in the following way, right? For an expression of natural numbers by 0, +, and ^ such that the out-most operation is ^ (if the expression is not 0), e.g. p^{p^p^0+p^p^0+p^0}, then the expression lies in a finite subset of the backgroud space N^{o(X^α)} in a natural way, right?
 * 1) If α = 0, then o(X^α) = 0..
 * 2) If α = β+X^γ, then o(X^α) = o(X^β)+o(X^X^γ).
 * 3) If α = X^γ, then o(X^α) = ω^{o(X^γ)}.