User blog comment:Ynought/Extended array hierarchy/@comment-35470197-20190117222337/@comment-35470197-20190121151908

For this purpose, you need to remember how FGH grows. In the definition of \(f_{\omega + 1}(n)\), we iterate \(f_{\omega}(n)\) \(n\)-times. In the definition of \(f_{\omega + 2}(n)\), we iterate \(f_{\omega + 1}(n)\) \(n\)-times. In the definition of \(f_{\omega + 3}(n)\), we iterate \(f_{\omega + 2}(n)\) \(n\)-times.

But instead, if you iterator \(f_{\omega}(n)\) \(f_{\omega+1}(n)\)-times, then the resulting function is bounded by \(f_{\omega+1}^2(n)\). If you iterator \(f_{\omega}(n)\) \(f_{\omega+1}^2(n)\), then the resulting function is bounded by \(f_{\omega+1}^3(n)\). So you could not reach \(f_{\omega+2}(n)\) in this way.

In order to grow in a faster way, you need to iterate the fastest function which has already been defined at the time. Many strong array notations follow such strategy. On the other hand, for example, your rule for \((a,b,\ldots,d)_{(e,f,\ldots,g)}\) just iterates \((a+d,b+d,\ldots,d+d)\), which is not the fastest one at the time, and the fatest function \((\bullet)_{(e,f,\ldots,g-1)}\) once.

Other rules are similar. Only the nesting rule uses the fastest function which has already been defined. That is why the growth rate is trapped before \(\omega \times 3\) in FGH.