Fast-growing hierarchy

A fast-growing hierarchy is a certain hierarchy mapping ordinals \(\alpha < \mu\) to functions \(f_\alpha: N \rightarrow N\). For large ordinals \(\alpha\), \(f_\alpha\) grows extremely fast (on par with BEAF arrays &mdash; see below).

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
 * \(f_\alpha(n) = f_{\alpha[n]}(n)\) iff \(\alpha\) 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:


 * \(\omega^{\alpha + 1}[n] = \omega^\alpha n \ldots\)
 * \(\omega^{\alpha}[n] = \omega^{\alpha[n]}\) iff \(\alpha\) 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 \(\epsilon_0[n + 1] = \omega^{\epsilon_0[n]}\)

The hierarchy can be extended into the Veblen function and up to the Feferman–Schütte ordinal \(\Gamma_0\). It has been defined for most recursive ordinals, but extending past \(\omega^\text{CK}_1\) is very difficult.

Approximations
Below are some functions in the Wainer hierarchy compared to arrow notation and BEAF. \(n\omega\) is here used as an abbreviation for \(\omega \times n = \underbrace{\omega + \cdots + \omega}_n\). \(m\) indicates any positive integer.


 * \(f_0(n) = n + 1\)
 * \(f_1(n) = f_0^n(n) = (((n + 1) + 1) + \cdots + 1) = n + n = 2n\)
 * \(f_2(n) = f_1^n(n) = 2(2(\ldots 2(2n))) = 2^n n > 2 \uparrow n\) (arrow notation)
 * \(f_3(n) > 2\uparrow\uparrow n\)
 * \(f_4(n) > 2\uparrow\uparrow\uparrow n\)
 * \(f_m(n) > 2\uparrow^{m-1} n\)
 * \(f_\omega(n) > 2\uparrow^{n-1} n = Ack(n)\) (Ackermann function)
 * \(f_{\omega+1}(n) > \lbrace n,n,1,2 \rbrace\) (BEAF)
 * \(f_{\omega+2}(n) > \lbrace n,n,2,2 \rbrace\)
 * \(f_{\omega+m}(n) > \lbrace n,n,m,2 \rbrace\)
 * \(f_{2\omega}(n) > \lbrace n,n,n,2 \rbrace\)
 * \(f_{2\omega}(n) > \lbrace n,n,n,3 \rbrace\)
 * \(f_{m\omega}(n) > \lbrace n,n,n,m \rbrace\)
 * \(f_{\omega^2}(n) > \lbrace n,n,n,n \rbrace\)
 * \(f_{\omega^3}(n) > \lbrace n,n,n,n,n \rbrace\)
 * \(f_{\omega^m}(n) > \lbrace n,m+2 (1) 2 \rbrace\)
 * \(f_{\omega^{\omega}}(n) > \lbrace n,n+2 (1) 2 \rbrace > \lbrace n,n (1) 2 \rbrace\)
 * \(f_{\omega^{\omega}+1}(n) > \lbrace n,n,2 (1) 2 \rbrace\)
 * \(f_{\omega^{\omega}+2}(n) > \lbrace n,n,3 (1) 2 \rbrace\)
 * \(f_{\omega^{\omega}+m}(n) > \lbrace n,n,m+1 (1) 2 \rbrace\)
 * \(f_{\omega^{\omega}+\omega}(n) > \lbrace n,n,n+1 (1) 2 \rbrace > \lbrace n,n,n (1) 2 \rbrace\)
 * \(f_{\omega^{\omega}+\omega+1}(n) > \lbrace n,n,1,2 (1) 2 \rbrace\)
 * \(f_{\omega^{\omega}+2 \omega}(n) > \lbrace n,n,n,2 (1) 2 \rbrace\)
 * \(f_{\omega^{\omega}+\omega^2}(n) > \lbrace n,n,n,n (1) 2 \rbrace\)
 * \(f_{{\omega^{\omega}}2}(n) > \lbrace n,n (1) 3 \rbrace\)
 * \(f_{{\omega^{\omega}}3}(n) > \lbrace n,n (1) 4 \rbrace\)
 * \(f_{{\omega^{\omega}}m}(n) > \lbrace n,n (1) m+1 \rbrace\)
 * \(f_{\omega^{\omega+1}}(n) > \lbrace n,n (1) n+1 \rbrace > \lbrace n,n (1) n \rbrace\)
 * \(f_{\omega^{\omega+2}}(n) > \lbrace n,n (1) n,n \rbrace\)
 * \(f_{\omega^{\omega+3}}(n) > \lbrace n,n,n (1) n,n,n \rbrace\)
 * \(f_{\omega^{\omega+m}}(n) > \lbrace n,m (1)(1) 2 \rbrace\)
 * \(f_{\omega^{2 \omega}}(n) > \lbrace n,n (1)(1) 2 \rbrace = \lbrace n,2 (2) 2 \rbrace\)
 * \(f_{\omega^{3 \omega}}(n) > \lbrace n,n (1)(1)(1) 2 \rbrace = \lbrace n,3 (2) 2 \rbrace\)
 * \(f_{\omega^{m \omega}}(n) > \lbrace n,m (2) 2 \rbrace\)
 * \(f_{\omega^{\omega^2}}(n) > \lbrace n,n (2) 2 \rbrace\)
 * \(f_{\omega^{\omega^3}}(n) > \lbrace n,n (3) 2 \rbrace\)
 * \(f_{\omega^{\omega^m}}(n) > \lbrace n,n (m) 2 \rbrace\)
 * \(f_{\omega^{\omega^\omega}}(n) > \lbrace n,n (n) 2 \rbrace = \lbrace n,n (0,1) 2 \rbrace\)
 * \(f_{^4{\omega}}(n) > \lbrace n,n ((1) 1) 2 \rbrace = n \uparrow\uparrow 3 \&\ n\), using array of operator.
 * \(f_{^5{\omega}}(n) > \lbrace n,n ((0,1) 1) 2 \rbrace = n \uparrow\uparrow 4 \&\ n\)
 * \(f_{^6{\omega}}(n) > \lbrace n,n (((1) 1) 1) 2 \rbrace = n \uparrow\uparrow 5 \&\ n\)
 * \(f_{^m{\omega}}(n) > n \uparrow\uparrow m-1 \&\ n\)
 * \(f_{\varepsilon_0}(n) > n \uparrow\uparrow n-1 \&\ n\)

Beyond \(\varepsilon_0\)

 * \(f_{\varepsilon_1}(n) \approx n \uparrow\uparrow n+1 \&\ n\)
 * \(f_{\varepsilon_2}(n) \approx n \uparrow\uparrow n+2 \&\ n\)
 * \(f_{\varepsilon_\omega}(n) \approx n \uparrow\uparrow 2n \&\ n\)
 * \(f_{\varepsilon_{2\omega}}(n) \approx n \uparrow\uparrow 3n \&\ n\)
 * \(f_{\varepsilon_{\omega^2}}(n) \approx n \uparrow\uparrow n^2 \&\ n\)
 * \(f_{\varepsilon_{\omega^\omega}}(n) \approx n \uparrow\uparrow n^n \&\ n\)
 * \(f_{\varepsilon_{\varepsilon_0}}(n) \approx n \uparrow\uparrow (n \uparrow\uparrow n) \&\ n\)
 * \(f_{ζ_0}(n) \approx n \uparrow\uparrow\uparrow n \&\ n\)
 * \(f_{η_0}(n) \approx n \uparrow^{4} n \&\ n\)
 * \(f_{\varphi(m,0)}(n) \approx n \uparrow^{m+1} n \&\ n > n \uparrow^{m} \&\ n\)
 * \(f_{\varphi(\omega,0)}(n) \approx n \uparrow^{n+1} n \&\ n > n \uparrow^{n} \&\ n = \lbrace n,2,1,2 \rbrace \&\ n\)
 * \(f_{\varphi(\omega+1,0)}(n) \approx \lbrace n,n,1,2 \rbrace \&\ n\)
 * \(f_{\varphi(\omega+2,0)}(n) \approx \lbrace n,n,2,2 \rbrace \&\ n\)
 * \(f_{\varphi(2\omega,0)}(n) \approx \lbrace n,n,n,2 \rbrace \&\ n\)
 * \(f_{\varphi(3\omega,0)}(n) \approx \lbrace n,n,n,3 \rbrace \&\ n\)
 * \(f_{\varphi(\omega^2,0)}(n) \approx \lbrace n,n,n,n \rbrace \&\ n\)
 * \(f_{\varphi(\omega^3,0)}(n) \approx \lbrace n,n,n,n,n \rbrace \&\ n\)
 * \(f_{\varphi(\omega^\omega,0)}(n) \approx \lbrace n,n (1) 2 \rbrace \&\ n\)
 * \(f_{\varphi(\omega^{2\omega},0)}(n) \approx \lbrace n,n (1)(1) 2 \rbrace \&\ n\)
 * \(f_{\varphi(\omega^{3\omega},0)}(n) \approx \lbrace n,n (1)(1)(1) 2 \rbrace \&\ n\)
 * \(f_{\varphi(\omega^{\omega^2},0)}(n) \approx \lbrace n,n (2) 2 \rbrace \&\ n\)
 * \(f_{\varphi(\omega^{\omega^3},0)}(n) \approx \lbrace n,n (3) 2 \rbrace \&\ n\)
 * \(f_{\varphi(\omega^{\omega^\omega},0)}(n) \approx \lbrace n,n (0,1) 2 \rbrace \&\ n\)
 * \(f_{\varphi(\omega^{\omega^{\omega^\omega}},0)}(n) \approx \lbrace n,n ((1) 1) 2 \rbrace \&\ n\)
 * \(f_{\varphi(\varepsilon_0,0)}(n) \approx n \uparrow\uparrow n \&\ n \&\ n\)
 * \(f_{\varphi(ζ_0,0)}(n) \approx n \uparrow\uparrow\uparrow n \&\ n \&\ n\)
 * \(f_{\varphi(η_0,0)}(n) \approx n \uparrow^{4} n \&\ n \&\ n\)
 * \(f_{Γ_0}(n) \approx \lbrace n,n / 2 \rbrace\)
 * \(f_{Γ_0+1}(n) \approx \lbrace n,n,2 / 2 \rbrace\)
 * \(f_{Γ_0+2}(n) \approx \lbrace n,n,3 / 2 \rbrace\)
 * \(f_{Γ_0+\omega}(n) \approx \lbrace n,n,n / 2 \rbrace\)
 * \(f_{Γ_0+\omega^2}(n) \approx \lbrace n,n,n,n / 2 \rbrace\)
 * \(f_{Γ_0+\omega^\omega}(n) \approx \lbrace n,n (1) 2 / 2 \rbrace\)
 * \(f_{Γ_0+\omega^{\omega^2}}(n) \approx \lbrace n,n (2) 2 / 2 \rbrace\)
 * \(f_{Γ_0 2}(n) \approx \lbrace n,n / 3 \rbrace\)
 * \(f_{Γ_0 3}(n) \approx \lbrace n,n / 4 \rbrace\)
 * \(f_{Γ_0\omega}(n) \approx \lbrace n,n / n \rbrace\)
 * \(f_{Γ_0{\omega^2}}(n) \approx \lbrace n,n / n,n \rbrace\)
 * \(f_{Γ_0{\omega^3}}(n) \approx \lbrace n,n / n,n,n \rbrace\)
 * \(f_{Γ_0{\omega^\omega}}(n) \approx \lbrace n,n / 1 (1) 2 \rbrace\)
 * \(f_{Γ_0{\omega^{\omega^2}}}(n) \approx \lbrace n,n / 1 (2) 2 \rbrace\)
 * \(f_{Γ_0{\varepsilon_0}}(n) \approx \lbrace n,n / n \uparrow \uparrow n \&\ n \rbrace\)
 * \(f_{{Γ_0}^2}(n) \approx \lbrace n,n / 1 / 2 \rbrace\)
 * \(f_{{Γ_0}^3}(n) \approx \lbrace n,n / 1 / 1 / 2 \rbrace\)
 * \(f_{{Γ_0}^\omega}(n) \approx \lbrace n,n (/1) 2 \rbrace\)
 * \(f_{{Γ_0}^{\omega+1}}(n) \approx \lbrace n,n (/1) 1 / 2 \rbrace\)
 * \(f_{{Γ_0}^{2\omega}}(n) \approx \lbrace n,n (/1)(/1) 2 \rbrace\)
 * \(f_{{Γ_0}^{3\omega}}(n) \approx \lbrace n,n (/1)(/1)(/1) 2 \rbrace\)
 * \(f_{{Γ_0}^{\omega^2}}(n) \approx \lbrace n,n (/2) 2 \rbrace\)
 * \(f_{{Γ_0}^{\omega^3}}(n) \approx \lbrace n,n (/3) 2 \rbrace\)
 * \(f_{{Γ_0}^{\omega^\omega}}(n) \approx \lbrace n,n (/0,1) 2 \rbrace\)
 * \(f_{{Γ_0}^{\omega^{\omega^\omega}}}(n) \approx \lbrace n,n (/(1) 1) 2 \rbrace\)
 * \(f_{{Γ_0}^{\varepsilon_0}}(n) \approx n \uparrow\uparrow n \&\&\ n\)
 * \(f_{{Γ_0}^{ζ_0}}(n) \approx n \uparrow\uparrow\uparrow n \&\&\ n\)
 * \(f_{{Γ_0}^{η_0}}(n) \approx n \uparrow^{4} n \&\&\ n\)
 * \(f_{{Γ_0}^{Γ_0}}(n) \approx \lbrace n,n / 2 \rbrace \&\&\ n\)
 * \(f_{{Γ_0}^{{Γ_0}^{Γ_0}}}(n) \approx \lbrace n,n / 2 \rbrace \&\&\ n \&\&\ n\)
 * \(f_(n) \approx \lbrace n,n // 2 \rbrace\)
 * \(f_(n) \approx \lbrace n,n /// 2 \rbrace\)
 * \(f_(n) \approx \lbrace n,n //// 2 \rbrace\)
 * \(f_(n) \approx \lbrace n,n ///.../// 2 \rbrace\), (m-2 /'s)
 * \(f_(n) \approx \lbrace n,n (1)/ 2 \rbrace\)
 * \(f_(n) \approx \lbrace n,n (2)/ 2 \rbrace\)
 * \(f_(n) \approx \lbrace n,n (3)/ 2 \rbrace\)
 * \(f_(n) \approx \lbrace n,n (m)/ 2 \rbrace\)
 * \(f_(n) \approx \lbrace n,n (0,1)/ 2 \rbrace\)
 * \(f_(n) \approx \lbrace n,n ((1) 1)/ 2 \rbrace\)
 * \(f_(n) \approx \lbrace L,n \uparrow\uparrow n \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,n \uparrow\uparrow\uparrow n \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,n \uparrow^{4} n \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,L \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,L+1 \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,L+2 \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,L+X \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,L+X \uparrow\uparrow X \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,2L \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,3L \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,XL \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,X \uparrow\uparrow X L \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L, \lbrace L,2 \rbrace \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L, \lbrace L,3 \rbrace \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L, \lbrace L,L \rbrace \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,X,2 \rbrace_{n,n}\)
 * \(f_(n) \approx \lbrace L,X,3 \rbrace_{n,n}\)
 * \(f_{{\varphi(\varphi(\omega,0),{Γ_0}})}(n) \approx \lbrace L,X,X \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0,1)}(n) \approx \lbrace L,L,L \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0+1,1)}(n) \approx \lbrace L,L,1,2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0+2,1)}(n) \approx \lbrace L,L,2,2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0+\omega,1)}(n) \approx \lbrace L,L,X,2 \rbrace_{n,n}\)
 * \(f_{\varphi(2Γ_0,1)}(n) \approx \lbrace L,L,L,2 \rbrace_{n,n}\)
 * \(f_{\varphi(3Γ_0,1)}(n) \approx \lbrace L,L,L,3 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^2,1)}(n) \approx \lbrace L,L,L,L \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^3,1)}(n) \approx \lbrace L,L,L,L,L \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^\omega,1)}(n) \approx \lbrace L,X (1) 2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{\varepsilon_0},1)}(n) \approx \lbrace L,X \uparrow\uparrow X (1) 2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{Γ_0},1)}(n) \approx \lbrace L,L (1) 2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{Γ_0+1},1)}(n) \approx \lbrace L,L (1) L \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{Γ_0+2},1)}(n) \approx \lbrace L,L (1) L,L \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{Γ_0+\omega},1)}(n) \approx \lbrace L,X (1)(1) 2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{2Γ_0},1)}(n) \approx \lbrace L,L (1)(1) 2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{3Γ_0},1)}(n) \approx \lbrace L,L (1)(1)(1) 2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{Γ_0^2},1)}(n) \approx \lbrace L,L (2) 2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{Γ_0^3},1)}(n) \approx \lbrace L,L (3) 2 \rbrace_{n,n}\)
 * \(f_{\varphi(Γ_0^{Γ_0^{Γ_0}},1)}(n) \approx \lbrace L,L (0,1) 2 \rbrace_{n,n}\)
 * \(f_{\varphi(\varepsilon_{Γ_0},1)}(n) \approx n \uparrow\uparrow n @ n\)
 * \(f_{\varphi(ζ_{Γ_0},1)}(n) \approx n \uparrow\uparrow\uparrow n @ n\)
 * \(f_{\varphi(η_{Γ_0},1)}(n) \approx n \uparrow^{4} n @ n\)
 * \(f_{\varphi(\varphi(Γ_0,1)),1)}(n) \approx \lbrace L,L,L \rbrace_{n,n} @ n\)
 * \(f_{Γ_1}(n) \approx \lbrace n,n \ 2 \rbrace = \lbrace L2,1 \rbrace_{n,n}\)
 * \(f_{Γ_2}(n) \approx \lbrace n,n | 2 \rbrace = \lbrace L3,1 \rbrace_{n,n}\)
 * \(f_{Γ_3}(n) \approx \lbrace n,n - 2 \rbrace = \lbrace L4,1 \rbrace_{n,n}\)
 * \(f_{Γ_\omega}(n) \approx \lbrace LX,1 \rbrace_{n,n}\)
 * \(f_{Γ_\omega^2}(n) \approx \lbrace LX,2 \rbrace_{n,n}\)
 * \(f_{Γ_{\varepsilon_0}}(n) \approx \lbrace LX \uparrow\uparrow X,1 \rbrace_{n,n}\)
 * \(f_{Г_{Γ_0}}(n) \approx \lbrace LL,1 \rbrace_{n,n}\)
 * \(f_{Г_{Γ_1}}(n) \approx \lbrace LL2,1 \rbrace_{n,n}\)
 * \(f_{Г_{Γ_{Γ_0}}}(n) \approx \lbrace LLL,1 \rbrace_{n,n}\)
 * \(f_{\varphi(1,1,0)}(n) \approx \lbrace (1)L,1 \rbrace_{n,n}\)
 * \(f_{\varphi(1,2,0)}(n) \approx \lbrace (2)L,1 \rbrace_{n,n}\)
 * \(f_{\varphi(1,3,0)}(n) \approx \lbrace (3)L,1 \rbrace_{n,n}\)
 * \(f_{\varphi(1,4,0)}(n) \approx \lbrace (4)L,1 \rbrace_{n,n}\)