Let μ be a large countable ordinal such to every limit ordinal α < μ there is assigned a fundamental sequence (a strictly increasing sequence of ordinals whose supremum is α). A accelerated-growing hierarchy of functions h_α: N → N, for α < μ, is then defined as follows:
\(h_0(n)= n\rightarrow^nn\)(Powerful Arrow Notation)
\(h_{\alpha+1}(n) = h_{\alpha}\alpha\rightarrow^\alpha\alpha(n\rightarrow^nn)\) until n = 0, in which case, refer to the top rule.(Powerful Arrow Notation)
\(h_{\alpha}(n) = h_{a[n\rightarrow^nn]}(n\rightarrow^nn)\)for some limit ordinal \(\alpha\)(Powerful Arrow Notation)
In order to understand the chained arrows, we must create a shorthand for repeated recursion, where I use the symbol \(\uparrow\):
\begin{eqnarray*} f_x \uparrow^n b(c) = \left\{ \begin{array}{ll} {f_x}^b(c) & (n = 1) \\ f_x(c) & (n > 1, b = 1) \\ f_x \uparrow^{n-1} (f_x \uparrow^{n-1} (f_x \uparrow^{n-1} (f_x \uparrow^n (b-1)(c))(c) & (n > 1, b > 1) \end{array} \right. \end{eqnarray*} Anything inside the [ ] will be repeated, i.e. \([3\uparrow]^{3}3 =3\uparrow3\uparrow3\uparrow3\) For a second level recoursion, I define a chained variant of the recursive arrow:
\(a \rightarrow b = a^{b}\)
\(f_x \rightarrow b \rightarrow c(d) = f_x\uparrow^cb \) (in which up-arrow notation is used)
\(f_x \rightarrow\ldots\rightarrow b \rightarrow 1(d) = f_x \rightarrow\ldots\rightarrow b(d)\) — When last entry is 1, it will be ignored.
\(f_x \rightarrow\ldots\rightarrow b \rightarrow 1 \rightarrow c(d) = f_x \rightarrow\ldots\rightarrow b(d)\)
\(f_x \rightarrow\ldots\rightarrow b \rightarrow (c + 1) \rightarrow (d + 1)(e) = f_x \rightarrow\ldots\rightarrow b \rightarrow (f_x \rightarrow\ldots\rightarrow b \rightarrow c \rightarrow (d + 1) ) \rightarrow d(e)\)
\(a\rightarrow^1 b = a^b\) \(f_x\rightarrow^x b(d) = f(a)\rightarrow^{x-1} f(a)\rightarrow^{x-1} f(a) \ldots \rightarrow^{x-1} f(a)(d)\) where there are \(b\rightarrow^{x-1}b\) copies of \(f(a)\), where \(f(a) = a\uparrow^aa\)
\(f_x\#\rightarrow^y 1(d) = f_x\#(d)\) (here, \(\#\) denotes an arbitrarily long part of the chain before the relevant terms)
\(f_x\#\rightarrow^y 1\rightarrow^z a(d) = f_x\#(d)\)
\(f_x\#\rightarrow^n a\rightarrow^1 b(d) = \#\rightarrow^n (\#\rightarrow^n (a-1)\rightarrow^1 b)\rightarrow^1 (b-1)(d)\)
\(f_x\#\rightarrow^n a\rightarrow^y b(d) = f_x\#\rightarrow^n a\rightarrow^{y-1} a\rightarrow^{y-1} a \ldots \rightarrow^{y-1} a(d)\) where there are \(b\) copies of \(a\)