User blog comment:Mh314159/Alpha numbers (and beyond)/@comment-35470197-20191007022858

Is there a definition of \(\alpha(a_i^j,k)\)? It is used in the definition of \(\alpha(a_b,x)\), but I could not find a rule to solve the superscript, because \(a_i\) is just regarded as a subscripted argument, which is not a number or a function.

Also, what does \(\alpha(a-1^m,x)\) mean? I guess that you do not intend \(a-1^m = a-(1^m) = a-1\) or \(a-1^m = (a-1)^m\). Since the superscript on a number usually means a power, you need to clarify the meaning and remove the ambiguity. Since your example shows \(0^m \neq 0\), I guess that you iterated the map \(x \mapsto \alpha(0,x)\). This iteration is ambiguous, because \(f(x,\ldots,y^m,z)\) does not mean the iteration of \(z \mapsto f(x,\ldots,y,z)\) in the usual mathematics.