User blog comment:DrCeasium/Continued hyperfactorial array notation/@comment-5529393-20130329065137

Sorry, I was incorrect regarding the strength of the notation. While it is true that the recursion has order type \(\omega^2\), the recursion step is basically

\(B_{\alpha+1}(n) = B_{\alpha}(B_{\alpha}(n)),

which is weaker than the recursion step in the fast-growing hierarchy. In fact, one step of the FGH is the equivalent of \(\omega\) steps in this hierarchy. So going \(\omega^2\) steps in this hierarchy is equivalent to going \(\omega\) steps in the FGH. Since the base function of hyperfactorial array notation is roughly the Ackermann function, or \(F_{\omega}(n)\), the strength of HAN is \(F_{\omega*2}(n)\), not \(F_{\omega^2}(n)\).