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 extremely fast. 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
 * \(f_\alpha(n) = f_{\alpha[n]}(n)\) 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:


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

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

The hierarchy can be extended into the and up to the  \(\Gamma_0\).

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.
 * \(m\omega\) is here used as a shorthand for \(\omega \times m = \underbrace{\omega + \cdots + \omega}_m\). In standard ordinal arithmetic, \(m \times \omega = \omega\).
 * \(^ab\) indicates tetration.
 * \(\uparrow\) indicates arrow notation.
 * \(\text{Ack}\) indicates the single-argument Ackermann function \(\text{Ack}(n, n)\).
 * \(\lbrace \rbrace\) indicates BEAF.

Up to \(\varepsilon_0\)
\begin{eqnarray*} 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 \\ 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_{2\omega}(n) &>& \lbrace n,n,n,2 \rbrace \\ f_{3\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 \\ 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 \end{eqnarray*}

Veblen hierarchy
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& n \uparrow\uparrow (n+1) \&\ n \\ f_{\varepsilon^{\varepsilon^{\varepsilon_0}_0}_0}(n) &\approx& n \uparrow\uparrow (n+2) \&\ n \\ f_{\varepsilon_1}(n) &\approx& n \uparrow\uparrow (2n) \&\ n \\ f_{\varepsilon_2}(n) &\approx& n \uparrow\uparrow (3n) \&\ n \\ f_{\varepsilon_\omega}(n) &\approx& n \uparrow\uparrow (n^2) \&\ n \\ f_{\varepsilon_{2\omega}}(n) &\approx& n \uparrow\uparrow (2n^2) \&\ n \\ f_{\varepsilon_{\omega^2}}(n) &\approx& n \uparrow\uparrow (n^3) \&\ 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 \\ \end{eqnarray*}

\(\Gamma_0\) and beyond
\begin{eqnarray*} 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 \underbrace{///...///}_{m - 2} 2 \rbrace \\ 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} \\ \end{eqnarray*} \begin{eqnarray*} 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 \backslash 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 (1)(1)L,1 \rbrace_{n,n} \\ f_{\varphi(1,3,0)}(n) &\approx& \lbrace (1)(1)(1)L,1 \rbrace_{n,n} \\ f_{\varphi(1,4,0)}(n) &\approx& \lbrace (1)(1)(1)(1)L,1 \rbrace_{n,n} \\ f_{\varphi(1,\omega,0)}(n) &\approx& \lbrace (2)L,1 \rbrace_{n,n} \\ f_{\varphi(1,\omega^2,0)}(n) &\approx& \lbrace (3)L,1 \rbrace_{n,n} \\ f_{\varphi(1,\omega^3,0)}(n) &\approx& \lbrace (4)L,1 \rbrace_{n,n} \\ f_{\varphi(1,\omega^m,0)}(n) &\approx& \lbrace (m+1)L,1 \rbrace_{n,n} \\ f_{\varphi(1,\omega^\omega,0)}(n) &\approx& \lbrace (0,1)L,1 \rbrace_{n,n} \\ f_{\varphi(1,\varepsilon_0,0)}(n) &\approx& \lbrace L \uparrow\uparrow L,1 \rbrace_{n,n} \\ f_{\varphi(1,ζ_0,0)}(n) &\approx& \lbrace L \uparrow\uparrow\uparrow L,1 \rbrace_{n,n} \\ f_{\varphi(1,η_0,0)}(n) &\approx& \lbrace L \uparrow^{4} L,1 \rbrace_{n,n} \\ f_{\varphi(1,Γ_0,0)}(n) &\approx& \lbrace L \&\ L,1 \rbrace_{n,n} \\ f_{\varphi(1,\varphi(1,Γ_0,0),0)}(n) &\approx& \lbrace L \&\ L \&\ L,1 \rbrace_{n,n} \\ f_{\varphi(2,0,0)}(n) &\approx& \lbrace L \&\ L \&\ L ... L \&\ L \&\ L,1 \rbrace_{n,n} = \lbrace L,L / 2 : L,1 \rbrace_{n,n} \\ f_{\varphi(2,0,1)}(n) &\approx& \lbrace L,L,L / 2 : L,1 \rbrace_{n,n} \\ f_{\varphi(2,0,2)}(n) &\approx& \lbrace L,L,L,L / 2 : L,1 \rbrace_{n,n} \end{eqnarray*}

Non-recursive ordinals
The smallest non-recursive ordinal is \(\omega^\text{CK}_1\), the. Here we have completely exhausted BEAF. In order, the functions are Rado's sigma function and 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_{\omega^\text{CK}_\omega}(n) &\approx& \text{Rayo}(n) \\ f_{\alpha}(n) &\approx& \Xi(n) \\ \end{eqnarray*}