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 Wainer hierarchy can be extended into the and up to the  \(\Gamma_0\). As long as fundamental sequences are defined, it is possible to extend FGH through all the recursive ordinals, or even all the countable ones.

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} \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*}