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. 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
A fast-growing hierarchy consists of an ordinal \(\mu\) and a fundamental sequence system \(S: \mu \cap \text{Lim} \rightarrow (\mathbb{N} \rightarrow \mu)\), 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.

For a fast-growing hierarchy to be useful to googologists, it is also expected to satisfy the property that for all \(\alpha < \beta < \mu\), \(f_\alpha\) is eventually dominated by \(f_\beta\).

Wainer hierarchy
Definitions of the fast-growing hierarchy and choices of fundamental sequence systems vary between authors, so it is generally problematic to speak of "the" fast-growing hierarchy. The most well-known FGH, however, is the Wainer hierarchy, which has \(\mu = \varepsilon_0 + 1\) and an FS system 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 \(1, \omega, \omega^2, \omega^3, \ldots\). When authors refer to "the fast-growing hierarchy" without clarification, the Wainer hierarchy is usually meant.

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.

\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*}

Buchholz hierarchy
In An Independence Result for (\(\Pi_1^1\)-CA) + BI, Wilfried Buchholz discusses an ordinal hierarchy where \(\mu = \psi_0\varepsilon_{\Omega_\omega + 1}\), where \(\psi\) is Buchholz's ordinal collapsing function and \(\psi_0\varepsilon_{\Omega_\omega + 1}\) is the TFB ordinal.

Other hierarchies
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.

Selecting fundamental sequences is not an easy problem, since some selections can lead to pathological hierarchies where \(\alpha < \beta\) does not necessarily imply \(f_\alpha <^* f_\beta\). Diana Schmidt's 1976 paper "Built-Up Systems of Fundamental Sequences and Hierarchies of Number-Theoretical Functions" discusses built-up fundamental sequence systems. For a fundamental sequence system to be built-up, for all \(\alpha\) and \(n\), \(\alpha[n+1]\) must be reducible to \(\alpha[n]\) by iterating the operations \(\beta \mapsto \beta - 1\) and \(\beta \mapsto \beta[0]\). A built-up fundamental sequence system guarantees that all the \(f_\alpha\) are monotonically increasing and that \(\alpha < \beta \Rightarrow f_\alpha <^* f_\beta\).

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.