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.

Definition
Definitions of the fast-growing hierarchy vary between authors, so it is generally problematic to speak of "the" fast-growing hierarchy.

A fast-growing hierarchy consists of an ordinal \(\mu\) and a fundamental sequence system \(S: \mu \cap \text{Lim} \rightarrow (\mathbb{N} \mapsto \mu)\), where \(S(\alpha)(n)\) is denoted \(\alpha[n]\). The semantics are as follows:


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

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

Wainer hierarchy
A well-known fast-growing hierarchy with \(\mu = \varepsilon_0 + 1\) is the Wainer hierarchy, defined as follows:


 * \(\omega[n] = n\)
 * \(\omega^{\alpha + 1}[n] = \omega^\alpha n\)
 * \(\omega^{\alpha}[n] = \omega^{\alpha[n]}\) 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 \(1, \omega, \omega^2, \omega^3, \ldots\)

We can easily replace the Wainer hierarchy with a different one, such as a hierarchy that works for much larger ordinals such as the small Veblen ordinal or the Takeuti-Feferman-Buchholz ordinal. As long as fundamental sequences are defined, it is possible to extend FGH up to an arbitrary countable ordinal. However, it is impossible to create an effective system that works for all countable ordinals, since a system of fundamental sequences up to \(\omega_1\) would be nonconstructive.

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 \(f_{\omega^\text{CK}_1}(n)\) is uncomputable and/or outgrows all computable functions, and if so, whether it also outgrows \(\Sigma(n)\) or if it's outgrown by it (or if they are comparable). Moreover, the answer to these questions depends on the choice of fundamental sequences for ordinals.