User blog comment:Mh314159/new YIP notation/@comment-39585023-20190705220451/@comment-35470197-20190707023925

I recommend to write the dependence of variables and functions in a local scope. For example, \(\{m\}_y(x)\) depends on \((N,\alpha,\beta)\), and hence it is better to denote it by \(\{m\}_y(x)_{N,\alpha,\beta}\) or something like that.

For a function \(g(\ldots)\), I denote by \(o(g(\ldots))\) the ordinal \(\gamma\) with fundamental sequences such that \(F_{\gamma}(\max \{\ldots\})\) approximately bounds \(g(\ldots)\). We have \begin{eqnarray*} o([a,b+1]) & \sim & 3 \times 2^{\omega} \times b + 1 o(\{0\}_0(x)_{N,1,\beta+1}) & \sim & \sup_{n \in \mathbb{N}} o([\underbrace{x,x,\ldots}_{N-2},n,\beta]) \\ o(\{0\}_0(x)_{N,\alpha+2,\beta+1}) & \sim & o([\underbrace{x,x,\ldots}_{N-2},\alpha+1,\beta+1]) \\ o(\{z+1\}_0(x)_{N,\alpha,\beta+1}\) & \sim & o(\{z\}_{z+1}(x)_{N,\alpha,\beta+1}) + 1 \\ o(\{z+1\}_{y+1}(x)_{N,\alpha,\beta+1}) & \sim & o(\{z+1\}_y(x)_{N,\alpha,\beta+1}) + 2 \ & \sim & o(\{z+1\}_0(x)_{N,\alpha,\beta+1}) + 2y \\ o(\{z+1\}_{y+1}(x)_{N,1,\beta+1}) & \sim & \left( \sup_{n \in \mathbb{N}} o([\underbrace{x,x,\ldots}_{N-2},n,\beta]) \right) + 2^{1+z} - 1 + 2y o(\{z+1\}_{y+1}(x)_{N,\alpha+2,\beta+1}) & \sim & o([\underbrace{x,x,\ldots}_{N-2},\alpha+1,\beta+1]) + 2^{1+z} - 1 + 2y o(m_{a,b,\ldota,1,\beta+1}) & \sim & \sup_{n \in \mathbb{N}} o([a,b,\ldots,n,\beta]) \\ o(m_{a,b,\ldota,\alpha+2,\beta+1}) & \sim & o([a,b,\ldots,\alpha+1,\beta+1]) \\ o([a,b,\ldots,1,\beta+1]) & \sim & \left( \sup_{n \in \mathbb{N}} o([\underbrace{n,n,\ldots}_{N-1},\beta]) \right) + 2^{\omega} \\ o([a,b,\ldots,\alpha+2,\beta+1]) & \sim & o([a,b,\ldots,\alpha+1,\beta+1]) + 2^{\omega} \begin{eqnarray*} and hence \end{eqnarray*} o([a,1,1]) & \sim & 2^{\omega} \times \omega \\ o([a,b+1,1]) & \sim & 2^{\omega} \times \omega + 2^{\omega} \times b \\ o([a,1,2]) & \sim & 2^{\omega} \times \omega \times 2 \\ o([a,b+1,2]) & \sim & 2^{\omega} \times \omega \times 2 + 2^{\omega} \times b \\ o([a,b+1,\beta + 1]) & \sim & 2^{\omega} \times (\omega \times (\beta + 1) + b) \\ o([a,b,1,1]) & \sim & 2^{\omega} \times \omega^2 \\ o([a,b,\alpha+1,\beta+1]) & \sim & 2^{\omega} \times (\omega^2 + \omega \times (\beta + 1) + \alpha)\\ o([a,b,\ldots,\alpha+1,\beta+1]) & \sim & 2^{\omega} \times (\omega^2 \times (N - 3) + \omega \times (\beta + 1) + \alpha)\\ o([n,\ldots,n]) & \sim & \omega^4. \end{eqnarray*} Therefore the limit of the notation is approximately bounded by \(\omega^4\) in FGH. I note that if you set \(\{m+1\}_0(x) = \{m\}_x^x(x)\) instead of \(\{m\}_m^m(x)\), then your system grows more similar to FGH. When you diagonalise the strongest function, the it is better to use a "main" variable instead of a constant in the local scope. For example, the "main" variable of \(\{m\}_y(x)\) is \(x\), and \(m\) and \(y\) are kind of constants because they are just used as indices of recursion.