Fast-growing hierarchy

A fast-growing hierarchy (FGH) is a certain hierarchy mapping ordinals \(\alpha < \mu\) to functions \(f_\alpha: N \rightarrow N\). For large ordinals \(\alpha\), \(f_\alpha\) grows very rapidly, and the growth rates it achieves are virtually limitless. Due to its simple and clear definition, as well as its origins in professional mathematics, FGH is a popular benchmark for large number functions, alongside BEAF (which is somewhat more esoteric).

Definitions
The most common definition:


 * \(f_0(n) = n + 1\)
 * \(f_{\alpha+1}(n) = f^n_\alpha(n)\), where \(f^n\) denotes function iteration
 * \(f_\alpha(n) = f_{\alpha[n]}(n)\) if and only if \(\alpha\) is a limit ordinal

\(\alpha[n]\) denotes the \(n\)th term of fundamental sequence assigned to ordinal \(\alpha\). Definitions of \(\alpha[n]\) can vary, giving different fast-growing hierarchies. For \(\alpha \leq \epsilon_0\), the so-called Wainer hierarchy uses the following definition:


 * \(\omega[n] = n\)
 * \(\omega^{\alpha + 1}[n] = \omega^\alpha n\)
 * \(\omega^{\alpha}[n] = \omega^{\alpha[n]}\) if and only if \(\alpha\) is a limit ordinal
 * \((\omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k})[n] = \omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k}[n]\) where \(\alpha_1 \geq \alpha_2 \geq \cdots \geq \alpha_{k - 1} \geq \alpha_k\)
 * \(\epsilon_0[0] = 0\) (alternatively \(1\)) and \(\epsilon_0[n + 1] = \omega^{\epsilon_0[n]}\)

For example, the fundamental sequence for \(\omega^\omega\) is \(1, \omega, \omega^2, \omega^3, \ldots\)

The Wainer hierarchy can be extended into the and up to the Feferman-Schütte ordinal \(\Gamma_0\). As long as fundamental sequences are defined, it is possible to extend FGH through the recursive ordinals up to \(\omega_1^\text{CK}\) (the Church-Kleene ordinal). Whether FGH exists for non-recursive ordinals &mdash; and if so, what it means &mdash; is an unresolved issue.

The general case where \(f_0\) is any increasing function forms a fast iteration hierarchy.

Approximations
Below are some functions in the Wainer hierarchy compared to other googological notations.

There are a few things to note:


 * Relationships denoted \(f_\alpha(n) > g(n)\) hold for sufficiently large \(n\), not necessarily all \(n\).
 * \(m\) indicates any positive integer.
 * \(^ab\) indicates tetration.
 * \(\uparrow\) indicates arrow notation.
 * \(\text{Ack}\) indicates the single-argument Ackermann function \(\text{Ack}(n, n)\).
 * \(\lbrace \rbrace\) indicates BEAF.
 * Expressions with \(X\) indicate structures.

Up to \(ε_0\)
\begin{eqnarray*} f_0(n) &=& n + 1 \\ f_1(n) &=& f_0^n(n) = ( \cdots ((n + 1) + 1) + \cdots + 1) = n + n = 2n \\ f_2(n) &=& f_1^n(n) = 2(2(\ldots 2(2n))) = 2^n n > 2 \uparrow n \\ f_3(n) &>& 2\uparrow\uparrow n \\ f_4(n) &>& 2\uparrow\uparrow\uparrow n \\ f_m(n) &>& 2\uparrow^{m-1} n \\ f_\omega(n) &>& 2\uparrow^{n-1} n = Ack(n) \\ f_{\omega+1}(n) &>& \lbrace n,n,1,2 \rbrace \\ f_{\omega+2}(n) &>& \lbrace n,n,2,2 \rbrace \\ f_{\omega+m}(n) &>& \lbrace n,n,m,2 \rbrace \\ f_{\omega2}(n) &>& \lbrace n,n,n,2 \rbrace \\ f_{\omega3}(n) &>& \lbrace n,n,n,3 \rbrace \\ f_{\omega m}(n) &>& \lbrace n,n,n,m \rbrace \\ f_{\omega^2}(n) &>& \lbrace n,n,n,n \rbrace \\ f_{\omega^3}(n) &>& \lbrace n,n,n,n,n \rbrace \\ f_{\omega^m}(n) &>& \lbrace n,m+2 (1) 2 \rbrace \\ f_{\omega^{\omega}}(n) &>& \lbrace n,n+2 (1) 2 \rbrace > \lbrace n,n (1) 2 \rbrace \\ f_{\omega^{\omega}+1}(n) &>& \lbrace n,n,2 (1) 2 \rbrace \\ f_{\omega^{\omega}+2}(n) &>& \lbrace n,n,3 (1) 2 \rbrace \\ f_{\omega^{\omega}+m}(n) &>& \lbrace n,n,m+1 (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega}(n) &>& \lbrace n,n,n+1 (1) 2 \rbrace > \lbrace n,n,n (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega+1}(n) &>& \lbrace n,n,1,2 (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega2}(n) &>& \lbrace n,n,n,2 (1) 2 \rbrace \\ f_{\omega^{\omega}+\omega^2}(n) &>& \lbrace n,n,n,n (1) 2 \rbrace \\ f_{{\omega^{\omega}}2}(n) &>& \lbrace n,n (1) 3 \rbrace \\ f_{{\omega^{\omega}}3}(n) &>& \lbrace n,n (1) 4 \rbrace \\ f_{{\omega^{\omega}}m}(n) &>& \lbrace n,n (1) m+1 \rbrace \\ f_{\omega^{\omega+1}}(n) &>& \lbrace n,n (1) n+1 \rbrace > \lbrace n,n (1) n \rbrace \\ f_{\omega^{\omega+2}}(n) &>& \lbrace n,n (1) n,n \rbrace \\ f_{\omega^{\omega+3}}(n) &>& \lbrace n,n,n (1) n,n,n \rbrace \\ f_{\omega^{\omega+m}}(n) &>& \lbrace n,m (1)(1) 2 \rbrace \\ f_{\omega^{\omega2}}(n) &>& \lbrace n,n (1)(1) 2 \rbrace = \lbrace n,2 (2) 2 \rbrace \\ f_{\omega^{\omega3}}(n) &>& \lbrace n,n (1)(1)(1) 2 \rbrace = \lbrace n,3 (2) 2 \rbrace \\ f_{\omega^{\omega m}}(n) &>& \lbrace n,m (2) 2 \rbrace \\ f_{\omega^{\omega^2}}(n) &>& \lbrace n,n (2) 2 \rbrace \\ f_{\omega^{\omega^3}}(n) &>& \lbrace n,n (3) 2 \rbrace \\ f_{\omega^{\omega^m}}(n) &>& \lbrace n,n (m) 2 \rbrace \\ f_{\omega^{\omega^\omega}}(n) &>& \lbrace n,n (n) 2 \rbrace = \lbrace n,n (0,1) 2 \rbrace \\ f_{^4{\omega}}(n) &>& \lbrace n,n ((1) 1) 2 \rbrace = n \uparrow\uparrow 3 \&\ n \\ f_{^5{\omega}}(n) &>& \lbrace n,n ((0,1) 1) 2 \rbrace = n \uparrow\uparrow 4 \&\ n \\ f_{^6{\omega}}(n) &>& \lbrace n,n (((1) 1) 1) 2 \rbrace = n \uparrow\uparrow 5 \&\ n \\ f_{^m{\omega}}(n) &>& X \uparrow\uparrow m-1 \&\ n \\ f_{\varepsilon_0}(n) &>& X \uparrow\uparrow n-1 \&\ n \end{eqnarray*}

Non-recursive ordinals
It is possible to define the fast-growing hierarchy for all recursive ordinals, and even for nonrecursive (but countable) ordinals. However, the definitions will necessarily be nonrecursive, making analysis far more complicated. For example, it is a nontrivial question whether \(F_{\omega^\text{CK}_1}(n)\) outgrows all computable functions, and if so, whether it also outgrows \(\Sigma(n)\), or is merely comparable to it (or whether \(\Sigma(n)\) outgrows it).