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).

The functions are usually defined 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.

\(\alpha[n]\) 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.

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.
 * \(m\alpha\) is here used as a shorthand for \(\alpha \times m = \underbrace{\alpha + \cdots + \alpha}_m\). In standard ordinal arithmetic, \(m \times \alpha \neq \alpha \times m\) if \(\alpha\) is a transfinite ordinal and \(1 < m < \omega\).
 * \(^ab\) 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*}

From \(\varepsilon_0\) to \(\Gamma_0\)
From here, \(\approx\) indicates a ballpark comparison, meaning that the two functions have similar rates of growth.

\begin{eqnarray*} f_{\varepsilon^{\varepsilon_0}_0}(n) &\approx& X \uparrow\uparrow (X+1) \&\ n \\ f_{\varepsilon^{\varepsilon^{\varepsilon_0}_0}_0}(n) &\approx& X \uparrow\uparrow (X+2) \&\ n \\ f_{\varepsilon_1}(n) &\approx& X \uparrow\uparrow (X*2) \&\ n \\ f_{\varepsilon_2}(n) &\approx& X \uparrow\uparrow (X*3) \&\ n \\ f_{\varepsilon_\omega}(n) &\approx& X \uparrow\uparrow (X^2) \&\ n \\ f_{\varepsilon_{\omega2}}(n) &\approx& X \uparrow\uparrow (2*X^2) \&\ n \\ f_{\varepsilon_{\omega^2}}(n) &\approx& X \uparrow\uparrow (X^3) \&\ n \\ f_{\varepsilon_{\omega^\omega}}(n) &\approx& X \uparrow\uparrow (X^X) \&\ n \\ f_{\varepsilon_{\varepsilon_0}}(n) &\approx& X \uparrow\uparrow (X \uparrow\uparrow X) \&\ n \\ f_{ζ_0}(n) &\approx& X \uparrow\uparrow\uparrow X \&\ n \\ f_{η_0}(n) &\approx& X \uparrow^{4} X \&\ n \\ f_{\varphi(m,0)}(n) &\approx& X \uparrow^{m+1} X \&\ n > n \uparrow^{m} \&\ n \\ f_{\varphi(\omega,0)}(n) &\approx& X \uparrow^{n+1} X \&\ n > X \uparrow^{X} \&\ X = \lbrace n,2,1,2 \rbrace \&\ n \\ f_{\varphi(\omega+1,0)}(n) &\approx& \lbrace X,X,X+1 \rbrace \&\ n \\ f_{\varphi(\omega+2,0)}(n) &\approx& \lbrace X,X,X+2 \rbrace \&\ n \\ f_{\varphi(\omega2,0)}(n) &\approx& \lbrace X,X,X*2 \rbrace \&\ n \\ f_{\varphi(\omega3,0)}(n) &\approx& \lbrace X,X,X*3 \rbrace \&\ n \\ f_{\varphi(\omega^2,0)}(n) &\approx& \lbrace X,X,X^2 \rbrace \&\ n \\ f_{\varphi(\omega^3,0)}(n) &\approx& \lbrace X,X,X^3 \rbrace \&\ n \\ f_{\varphi(\omega^\omega,0)}(n) &\approx& \lbrace X,X,X^X \rbrace \&\ n \\ f_{\varphi(\omega^{\omega2},0)}(n) &\approx& \lbrace X,X,X^{X*2} \rbrace \&\ n \\ f_{\varphi(\omega^{\omega3},0)}(n) &\approx& \lbrace X,X,X^{X*3} \rbrace \&\ n \\ f_{\varphi(\omega^{\omega^2},0)}(n) &\approx& \lbrace X,X,X^{X^2} \rbrace \&\ n \\ f_{\varphi(\omega^{\omega^3},0)}(n) &\approx& \lbrace X,X,X^{X^3} \rbrace \&\ n \\ f_{\varphi(\omega^{\omega^\omega},0)}(n) &\approx& \lbrace X,X,X^{X^X} \rbrace \&\ n \\ f_{\varphi(\omega^{\omega^{\omega^\omega}},0)}(n) &\approx& \lbrace X,X,X^{X^{X^X}} \rbrace \&\ n \\ f_{\varphi(\varepsilon_0,0)}(n) &\approx& \lbrace X,X,X \uparrow\uparrow X \rbrace \&\ n \\ f_{\varphi(ζ_0,0)}(n) &\approx& \lbrace X,X,X \uparrow\uparrow\uparrow X \rbrace \&\ n \\ f_{\varphi(η_0,0)}(n) &\approx& \lbrace X,X,X \uparrow^{4} X \rbrace \&\ n \\ f_{\varphi(\varphi(\omega,0),0)}(n) &\approx& \lbrace X,X,X \uparrow^{X} X \rbrace \&\ n = \lbrace X,X, \lbrace X,X,X \rbrace\rbrace \&\ n \\ f_{\varphi(\varphi(\varphi(\omega,0),0),0)}(n) &\approx& \{X,4,1,2\} \&\ n \\ \end{eqnarray*}

From \(\Gamma_0\) to \(\vartheta(\Omega^\Omega)\)
\begin{eqnarray*} f_{Γ_0}(n) &\approx& \lbrace X,X,1,2 \rbrace \&\ n \\ f_{\varphi(Γ_0,1)}(n) &\approx& \lbrace X,X+1,1,2 \rbrace \&\ n \\ f_{Γ_1}(n) &\approx& \lbrace X,X*2,1,2 \rbrace \&\ n \\ f_{Γ_2}(n) &\approx& \lbrace X,X*3,1,2 \rbrace \&\ n \\ f_{Г_\omega}(n) &\approx& \lbrace X,X^2,1,2 \rbrace \&\ n \\ f_{Г_{Г_0}}(n) &\approx& \lbrace X,\lbrace X,X,1,2 \rbrace,1,2 \rbrace \&\ n \\ f_{\varphi(1,1,0)}(n) &\approx& \lbrace X,X,2,2 \rbrace \&\ n \\ f_{\varphi(1,2,0)}(n) &\approx& \lbrace X,X,3,2 \rbrace \&\ n \\ f_{\varphi(1,\omega,0)}(n) &\approx& \lbrace X,X,X,2 \rbrace \&\ n \\ f_{\varphi(1,\varphi(1,0,0),0)}(n) &\approx& \lbrace X,X,\lbrace X,X,1,2 \rbrace,2 \rbrace \&\ n \\ f_{\varphi(2,0,0)}(n) &\approx& \lbrace X,X,1,3 \rbrace \&\ n \\ f_{\varphi(3,0,0)}(n) &\approx& \lbrace X,X,1,4 \rbrace \&\ n \\ f_{\varphi(\omega,0,0)}(n) &\approx& \lbrace X,X,1,X \rbrace \&\ n \\ f_{\varphi(\varphi(1,0,0),0,0)}(n) &\approx& \lbrace X,X,1,\lbrace X,X,1,2 \rbrace\rbrace \&\ n \\ f_{\varphi(1,0,0,0)}(n) &\approx& \lbrace X,X,1,1,2 \rbrace \&\ n \\ f_{\varphi(2,0,0,0)}(n) &\approx& \lbrace X,X,1,1,3 \rbrace \&\ n \\ f_{\varphi(1,0,0,0,0)}(n) &\approx& \lbrace X,X,1,1,1,2 \rbrace \&\ n \\ f_{\vartheta(\Omega^\omega)}(n) &\approx& \lbrace X,X (1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega+1})}(n) &\approx& \lbrace X,X (1) X \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega+2})}(n) &\approx& \lbrace X,X (1) X,X \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega2})}(n) &\approx& \lbrace X,X (1)(1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega3})}(n) &\approx& \lbrace X,X (1)(1)(1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega^2})}(n) &\approx& \lbrace X,X (2) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega^3})}(n) &\approx& \lbrace X,X (3) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega^\omega})}(n) &\approx& \lbrace X,X (0,1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\omega^{\omega^\omega}})}(n) &\approx& \lbrace X,X ((1) 1) 2 \rbrace \&\ n \\ f_{\vartheta(\Omega^{\vartheta(1)})}(n) &\approx& X \uparrow\uparrow X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(2)})}(n) &\approx& X \uparrow\uparrow\uparrow X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(3)})}(n) &\approx& X \uparrow^{4} X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\omega)})}(n) &\approx& X \uparrow^{X} X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\Omega)})}(n) &\approx& \lbrace X,X,1,2 \rbrace \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\Omega^\omega)})}(n) &\approx& \lbrace X,X (1) 2 \rbrace \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(1)})})}(n) &\approx& X \uparrow\uparrow X \&\ X \&\ X \&\ n \\ f_{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(1)})})})}(n) &\approx& X \uparrow\uparrow X \&\ X \&\ X \&\ X \&\ n \\ \end{eqnarray*}

Beyond \(\vartheta(\Omega^\Omega)\)
\begin{eqnarray*} f_{\vartheta(\Omega^\Omega)}(n) &\approx& \lbrace n,n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega,1)}(n) &\approx& \lbrace n,2n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega,\omega)}(n) &\approx& \lbrace n,n^2 / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega,\omega^\omega)}(n) &\approx& \lbrace n,n^n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega,\varepsilon_0)}(n) &\approx& \lbrace n,n \uparrow\uparrow n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega,\zeta_0)}(n) &\approx& \lbrace n,n \uparrow\uparrow\uparrow n / 2 \rbrace \\ f_{\vartheta(\Omega^\Omega,\varphi(\omega,0))}(n) &\approx& \lbrace n,\lbrace n,n,n \rbrace / 2 \rbrace \end{eqnarray*}

Non-recursive ordinals
It is possible to define the fast-growing hierarchy for all recursive ordinals, and even for nonrecursive 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 therefore is comparable in growth rate to \(\Sigma(n)\).

The smallest non-recursive ordinal is \(\omega^\text{CK}_1\), the. BEAF, being a computable function, is completely exhausted by now. In order, the functions are Rado's sigma function, its higher-order cousins, Rayo's function, and the xi function. Here \(\alpha\) is the first ordinal such that \(\omega^\text{CK}_{\alpha} = \alpha\).

\begin{eqnarray*} f_{\omega^\text{CK}_1}(n) &\approx& \Sigma(n) \\ f_{\omega^\text{CK}_2}(n) &\approx& \Sigma_2(n) \\ f_{\omega^\text{CK}_3}(n) &\approx& \Sigma_3(n) \\ f_{\omega^\text{CK}_m}(n) &\approx& \Sigma_m(n) \\ f_{\alpha}(n) &\approx& \Xi(n) \\ \end{eqnarray*}