User blog comment:Mh314159/FOX notation updated/@comment-35470197-20191213010937

> [n]‹n›(x) = is recursed using the same rule as f‹n›(x), treating [n] as f.

It is quite ambiguous. For example, since n](x) is defined only for the case n=m, the recursion [2](0) = [1]<2>(2) does not make sense.

In order to avoid such ambiguity, it is good to define a map (i.e. correspondence) F which assigns f[n](x) to each h(x):N→N and n∈N, instead of specialising the definition for x+1. (Strictly speaking, "replacing blah-blah in the definition by …" is not formalised well.) Then you can safely define f[n] = F(x+1,[n]) and [n](x) = F([n],m). If you want to "repeat the same constrution", then formalising it as a map is suitable.