User blog comment:Mh314159/new YIP notation/@comment-39585023-20190705220451/@comment-35470197-20190706011652

The \(\{0\}\{x)\) seems to be a typo of \(\{0\}(x)\). We have \begin{eqnarray*} f_a(b) & sim & F_{2a}(b) \\ [a] & \sim & F_{2(1 + \omega) + 1}(a) \\ \{0\}_0(x)(b=1) & sim & F_{2(1 + \omega) + 1}^2(x) \\ \{0\}_y(x)(b=1) & \sim & F_{2(1 + \omega) + 1 + 2y}(x) \\ \{1\}_0(x)(b=1) & \sim & F_{2(1 + \omega) + 3}(x) \\ \{1\}_y(x)(b=1) & \sim & F_{2(1 + \omega) + 3 + 2y}(x) \\ \{2\}_0(x)(b=1) & \sim & F_{2(1 + \omega) + 7}^2(x) \\ \{2\}_y(x)(b=1) & \sim & F_{2(1 + \omega) + 7 + 2y}(x) \\ \{z\}_y(x)(b=1) & \sim & F_{2(1 + \omega) + 2^{1+z} - 1 + 2y}(x) \\ m(a,1) & \sim & F_{2(1 + \omega) + 1}^2(a) \\ [a,1] & \sim & F_{3 \times 2^{\omega} + 1}(F_{2(1 + \omega) + 1}^2(a)) \\ \{0\}_0(x)(b=2) & sim & F_{3 \times 2^{\omega} + 1}^2(x) \\ \{z\}_y(x)(b=2) & \sim & F_{3 \times 2^{\omega} + 2^{1+z} - 1 + 2y}(x) \\ m(a,2) & \sim & F_{3 \times 2^{\omega} + 1}^2(a) \\ [a,2] & \sim & F_{3 \times 2^{\omega} \times 2 + 1}(F_{3 \times 2^{\omega} + 1}^2(a)) \\ [a,b] & \sim & F_{3 \times 2^{\omega} \times b + 1}(F_{3 \times 2^{\omega} \times (b-1) + 1}^2(a)) \\ [n,n] & \sim & F_{\omega^2}(n) \end{eqnarray*} Therefore the growth rate is roughly approximated by \(\omega^2\) in FGH. Good!

> I imagine I could also have made an expression more complex than [[a,b-1],b-1] if I used more symbols and definitions, but isn't that always the case?

Right. You can always strengthen your notation by adding new symbols, but it does not necessarily effect the ordinal in FGH. It depends on how you add.

> Isn't the idea to be as strong as you can but also be as compact as you can?

Concering th strength, your current notation is of level 7 in my googological ruler. It is better than average \(2\)-ary notations. (To be fair, your notation includes the \(3\)-ary system \(\{z\}_y(x)\), though.) My best \(1\)-ary elementary notation (without including other systems) is of level 14, but it is not fair to compare my systems because I know higher recursions using large ordinals. The compactness is also sufficiently good! Please read back your first notation. Then you will find how compact your current notation is.