User blog comment:Mh314159/new YIP notation/@comment-39585023-20190714235606/@comment-35470197-20190718134859

The rules "\(n = [a-1], [a] = g_n^n(x) \)" are a little confusing, because the dependency of these two variables are not declared. In order to avoid such an ambiguity, we declare variables by saying like "For a positive integer \(a\), we put \(n = [a-1])" or "For a positive integer \(n\), take a unique positive integer \(a\) such that \(n = [a-1]\)". I guess that you intend the first sentence, because the single bracket does not seem to be surjective. The problem on the dependency is a little more serious for Rule set 2. May be the rule for \(Y\) should be placed after the rule for \(\underline{0}_0(x)\), because you use \(\underline{0}_0(x)\) only for the case where the first entry is positive.

Also, the sentence "If \(Y = \cdots\)" should mean a case classification with a condition \(Y = \cdots\), but \(Y\) is undefined. Therefore I guess that this is a defining formula of \(Y\). In order to distinguish an equation displaying a condition and an equation defining a value, it is good to use the definition symbol \(:=\) or \(\stackrel{\textrm{def}}{=}\).

For a function \(\varphi(x)\), I denote by \(O(\varphi(x))\) the ordinal in FGH corresponding to \(\varphi(x)\). Here is the analysis for the first version, i.e. the one without proposed new Universal Relationships: \begin{eqnarray*} f_a(b) & \sim & F_{2a}(b) \\ f_x(x) & \sim & F_{2 \omega}(x) \\ O(\underline{m+1}_y(x)) & \sim & O(\underline{0}_0(x)) + 2 \omega + (1 + 2 \omega) \times (m-1) + 2y \\ O(g_0(x)) & \sim & O(\underline{0}_0(x)) + 2 \omega + (1 + 2 \omega) \times \omega \\ O(g_x^x(x)) & \sim & O(\underline{0}_0(x)) + 2 \omega + (1 + 2 \omega) \times \omega + 2 \\ [a] & \sim & F_{2 \omega \times 2 + (1 + 2 \omega) \times \omega + 2}^a(4) \\ O([\underbrace{x,x\ldots,x}_{N+1},\beta+1]) & \sim & \max \{O([\underbrace{x,\ldots,x}_N,\beta+1]),O([\underbrace{x,x\ldots,x}_{N+1},\beta]) + 2 \omega + (1 + 2 \omega) \times \omega + 3\} \\ O([\underbrace{x,\ldots,x}_{x}]) & \sim & 2 \omega + (2 \omega + (1 + 2 \omega) \times \omega + 3) \times \omega^2 \sim \omega^4 \end{eqnarray*} So the limit of this notation is \(\omega^4\) in FGH, if I am correct.