User blog comment:Mh314159/FOX notation/@comment-35470197-20191204015153/@comment-39585023-20191204233122

I understand that changing the first term in a three term index has a small effect on the growth rate. I don't think I agree with "f(x) is the strongest below f(x)." I think that the final term in a three term string has the most effect on the growth rate, and therefore f(x) is not below f(x). For example, let's compare f<1,0,2>(2)  to f<2,0,1>(2). By zero sub, f<1,0,2>(2) = f<1,(f<1,2,1>(2)),1>(2)  whereas f<2,0,1>(2) = f<2,(f<2,2>(2))>(2)  The former is a three term index and the latter is only a two term index. And there is no definition of the form f(x) = fx(x) because before generating the subscript the zero sub would occur. Subscripts can iterate indexes containing zeroes, but indexes containing zeroes cannot recurse to subscripted expressions. And in fact, f(x) can only arise by zero sub into f<0,1,c+1>(x)  where a+1 is a function of x. So as currently defined, f(x) = f(x)),c>(x) which then recurses to f(x)),c>xp(x). If we continue to decrement terms in strings that already have zeroes, it becomes unclear to me when to do zero substitution and therefore unclear how the recursion terminates. I hope you understand my logic here and I hope it helps you understand the system better. And if you have suggestions for improvement I'm interested in hearing them.