User blog comment:Mh314159/A new approach/@comment-35470197-20190702151646/@comment-35470197-20190703024720

The rule set for the row of brackets with strings does not effect so well against the ordinal in FGH, because it just gives a single recursion of previously defined function of a fixed level.

For example, in FGH, \(F_{\alpha + 1}(n)\) for an ordinal \(\alpha\) is the \(n\)-folded recursion of the strongest function before it, i.e. \(F_{\alpha}\). This is why FGH is so strong. On the other hand, if \(F_{\alpha + 1}(n)\) were just the \(F_{\alpha}(n)\)-folded recursion of a weaker function such as \(F_{\beta}\) satisfying \(\beta < \alpha\), then the resulting \(F_{\alpha + 1}\) would not not so significantly larger than \(F_{\alpha}(n)\).

The rule set for the row of brackets with strings is kind of the second type, while the rule set for [a,b] is kind of the first type. That is why it does not effect so well against the ordinal in FGH.