User blog comment:Mh314159/Natural number recursion -- first 6 rule sets/@comment-35470197-20191031004425

> Anyway, it doesn't help me with my natural number recursions because I cannot find the clear conversions between the FGH and my recursions

If you understand recursions below Veblen, it is sufficiently great, because then you are ready for creating a notation of ω^ω^ω level. I note that Veblen hierarchy is much more difficult than ω-powers, and hence it is natural. I n order to study in a step-by-step way, you do not have to reach it. I just expected for you to understand up to ω^ω so that you can analyse notations of your current level in order to create new one. Anyway, good effort.

> A‹S› and A‹T› are functions for all sequences S and T as defined below

What is informal is that you are not declaring the domain of A‹S›. Are you considering A‹S› for all sequences S? Or is it restricted by the condition "S = sequence of one or more nonzero terms"? ("As defined below" is incorrect, because declaring a condition is not a definition, which characterises a single unique object.)


 * If you consider "all sequences", then there is no rule applicable to a sequence including two non-trailing zeros.
 * If you restrict it by the condition, then the rules for zeros nver applicable.

I guess that the domain is the set of all non-empty sequences S which include at most one zero. If it is correct, then it is better to declare it.

For example, you can start rule 2 by writing the following:
 * I define a function A‹S› for each non-empty sequence S including at mmost one zero, which assigns a natural number to each natural number.
 * When S contains no zero, ... where T is defined below.
 * When S contains a trailling zero, ...
 * When S contains a non-trailling zero, ...

I will read the actual definition later.