User blog comment:Mh314159/Natural number recursion - first 4 rule sets/@comment-35470197-20191019143654/@comment-39585023-20191020004455

Well, now I'm a little discouraged because some of this is beyond me. I will say that I am calling A the "class" and S the "index" but intend A⟨S⟩ to be a function and x to be the argument of the function. The domain of S and x has been stated and the range of A⟨S⟩(x) is a natural number by recursion to A⟨T⟩(x) as defined. The superscript is functional recursion, so that A⟨S⟩m(x) = A⟨S⟩(A⟨S⟩m-1(x)) and A⟨S⟩1(x) = A⟨S⟩(x). I thought this was standard, but perhaps it is not because I needed to specify A⟨S⟩ as a function. And x can indeed be zero, because there is a definition for A⟨S⟩(0), but I will add that S is made up of nonzero natural numbers. Does this address what you were critiquing? I'm not sure I understood it completely because of my minimum training in formal math.