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The slow-growing hierarchy (SGH) is a certain hierarchy mapping ordinals \(\alpha\) to functions \(g_\alpha: \mathbb{N} \rightarrow \mathbb{N}\). Like its name suggests, it grows much slower than its cousins the fast-growing hierarchy and the Hardy hierarchy.

The functions are defined as follows:

  • \(g_0(n) = 0\)
  • \(g_{\alpha+1}(n) = g_\alpha(n)+1\)
  • \(g_\alpha(n) = g_{\alpha[n]}(n)\) when \(\alpha\) is a limit ordinal

\(\alpha[n]\) denotes the \(n\)th term of the fundamental sequence assigned to ordinal \(\alpha\). Definitions of \(\alpha[n]\) can vary, giving different slow-growing hierarchies. One such hierarchy is the Wainer hierarchy, which is explained in the article for fast-growing hierarchy.

For small ordinals, SGH is nowhere close to FGH. \(g_{\varepsilon_0}(n)\) only reaches the level of \(f_3(n)\), and SGH does not reach \(f_{\varepsilon_0}(n)\) until the Bachmann-Howard ordinal. Unlike its relatives, SGH is extremely sensitive to the definitions of fundamental sequences: using the most "natural" version SGH catches up FGH at \(\psi_0(\Omega_\omega)\). Other systems of fundamental sequences, however, have this 'catching ordinal' at \(\varepsilon_0\), \(\vartheta(\Omega^\Omega)\), or even beyond \(\psi(\psi_I(0))\) in some cases.

To googologists, SGH is not quite as useful as FGH. It grows the slowest of all the ordinal hierarchies, so it may be the best suited to stratify the growth rates of functions, if the fundamental sequences are properly specified. However, it has been theorized that it may be useful in creating fast growing functions in a similar manner to Goodstein sequences.

Functions Edit

Below is the list of comparisons between SGH and other googological notations.

Note: A specific system of fundamental sequences was used to obtain these approximations. A different system will likely produce wildly different values, so take care.

Up to \(\Gamma_0\) Edit

\(g_0(n) = 0\)

\(g_1(n) = 1\)

\(g_2(n) = 2\)

\(g_m(n) = m\)

\(g_\omega(n) = n\)

\(g_{\omega^{\omega}}(n) = n^n\)

\(g_{\omega^{\omega^{\omega}}}(n) = n^{n^{n}}\)

\(g_{\varepsilon_0}(n) = n \uparrow\uparrow n\) (See epsilon zero)

\(g_{\varepsilon_1}(n) \approx n \uparrow\uparrow (2n)\)

\(g_{\varepsilon_2}(n) \approx n \uparrow\uparrow (3n)\)

\(g_{\varepsilon_{\omega}}(n) \approx n \uparrow\uparrow (n^2)\)

\(g_{\varepsilon_{\omega^2}}(n) \approx n \uparrow\uparrow (n^3)\)

\(g_{\varepsilon_{\omega^3}}(n) \approx n \uparrow\uparrow (n^4)\)

\(g_{\varepsilon_{\omega^{\omega}}}(n) \approx n \uparrow\uparrow (n^n)\)

\(g_{\varepsilon_{\varepsilon_0}}(n) \approx n \uparrow\uparrow (n \uparrow\uparrow n)\)

\(g_{\zeta_0}(n) \approx n \uparrow\uparrow\uparrow n\)

\(g_{\varepsilon_{\zeta_0+1}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow n\)

\(g_{\varepsilon_{\zeta_0+2}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (2n)\)

\(g_{\varepsilon_{\zeta_0 2}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n) \approx n \uparrow\uparrow\uparrow (n+1)\)

\(g_{\varepsilon_{\zeta_0 3}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (2(n \uparrow\uparrow\uparrow n))\)

\(g_{\varepsilon_{\zeta_0 4}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (3(n \uparrow\uparrow\uparrow n))\)

\(g_{\varepsilon_{\zeta_0 \omega}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n(n \uparrow\uparrow\uparrow n))\)

\(g_{\varepsilon_{\zeta_0^2}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ({(n \uparrow\uparrow\uparrow n)}^2)\)

\(g_{\varepsilon_{\zeta_0^{\zeta_0}}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ({(n \uparrow\uparrow\uparrow n)}^{n \uparrow\uparrow\uparrow n})\)

\(g_{\varepsilon_{\varepsilon_{\zeta_0+1}}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow n)\)

\(g_{\varepsilon_{\varepsilon_{\zeta_0 2}}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n)) \approx n \uparrow\uparrow\uparrow (n+2)\)

\(g_{\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0+1}}}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow n))\)

\(g_{\zeta_1}(n) \approx n \uparrow\uparrow\uparrow 2n\)

\(g_{\zeta_2}(n) \approx n \uparrow\uparrow\uparrow 3n\)

\(g_{\zeta_\omega}(n) \approx n \uparrow\uparrow\uparrow n^2\)

\(g_{\zeta_{\omega^\omega}}(n) \approx n \uparrow\uparrow\uparrow n^n\)

\(g_{\zeta_{\varepsilon_0}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow n)\)

\(g_{\zeta_{\varepsilon_{\varepsilon_0}}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow (n \uparrow\uparrow n))\)

\(g_{\zeta_{\zeta_0}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow\uparrow n)\)

\(g_{\zeta_{\zeta_{\zeta_0}}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow\uparrow (n \uparrow\uparrow\uparrow n\)))

\(g_{\eta_0}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\)

\(g_{\varepsilon_{\eta_0+1}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow n\)

\(g_{\varepsilon_{\eta_0+2}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow 2n\)

\(g_{\varepsilon_{\eta_0+\omega}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow n^2\)

\(g_{\varepsilon_{\eta_0+\omega^\omega}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow n^n\)

\(g_{\varepsilon_{\eta_0+\varepsilon_0}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow n)\)

\(g_{\varepsilon_{\eta_0+\varepsilon_{\varepsilon_0}}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow n \uparrow\uparrow n)\)

\(g_{\varepsilon_{\eta_0+\zeta_0}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n)\)

\(g_{\varepsilon_{\eta_0+\zeta_1}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow 2n)\)

\(g_{\varepsilon_{\eta_0+\zeta_\omega}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n^2)\)

\(g_{\varepsilon_{\eta_0+\zeta_{\omega^\omega}}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n^n)\)

\(g_{\varepsilon_{\eta_0+\zeta_{\zeta_0}}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n \uparrow\uparrow\uparrow n)\)

\(g_{\varepsilon_{\eta_0 2}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow\uparrow n)\)

\(g_{\varepsilon_{\eta_0 3}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow 2(n \uparrow\uparrow\uparrow\uparrow n)\)

\(g_{\varepsilon_{\eta_0 \omega}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow n-1(n \uparrow\uparrow\uparrow\uparrow n)\)

\(g_{\zeta_{\eta_0+1}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow\uparrow n\)

\(g_{\varphi(4,0)}(n) \approx n \uparrow\uparrow\uparrow\uparrow\uparrow n\)

\(g_{\varphi(5,0)}(n) \approx n \uparrow\uparrow\uparrow\uparrow\uparrow\uparrow n\)

\(g_{\varphi(\omega,0)}(n) \approx \{n,n,n+1\}\)

\(g_{\varphi(\omega^\omega,0)}(n) \approx \{n,n,n^n+1\}\)

\(g_{\varphi(\varepsilon_0,0)}(n) \approx \{n,n,n \uparrow\uparrow n+1\}\)

\(g_{\varphi(\zeta_0,0)}(n) \approx \{n,n,n \uparrow\uparrow\uparrow n+1\}\)

\(g_{\varphi(\eta_0,0)}(n) \approx \{n,n,n \uparrow\uparrow\uparrow\uparrow n+1\}\)

\(g_{\varphi(\varphi(\omega,0),0)}(n) \approx \{n,n,\{n,n,n+1\}+1\}\)

\(g_{\varphi(\varphi(\varphi(\omega,0),0),0)}(n) \approx \{n,n,\{n,n,\{n,n,n+1\}+1\}+1\}\)

From \(\Gamma_0\) to \(\vartheta(\Omega^\omega)\) Edit

\(g_{\Gamma_0}(n) \approx \{n,n,1,2\}\)

\(g_{\varphi(\Gamma_0,1)}(n) \approx \{n,n+1,1,2\}\)

\(g_{\varphi(\varphi(\Gamma_0,1),0)}(n) \approx \{n,n+2,1,2\}\)

\(g_{\Gamma_1}(n) \approx \{n,2n,1,2\}\)

\(g_{\Gamma_2}(n) \approx \{n,3n,1,2\}\)

\(g_{\Gamma_\omega}(n) \approx \{n,(n+1)n,1,2\}\)

\(g_{\Gamma_{\omega^2}}(n) \approx \{n,(n^2+1)n,1,2\}\)

\(g_{\Gamma_{\omega^\omega}}(n) \approx \{n,(n^{n-1}+1)n,1,2\}\)

\(g_{\Gamma_{\omega^{\omega^\omega}}}(n) \approx \{n,(n^{n^n-1}+1)n,1,2\}\)

\(g_{\Gamma_{\varepsilon_0}}(n) \approx \{n,n \uparrow\uparrow n,1,2\}\)

\(g_{\Gamma_{\zeta_0}}(n) \approx \{n,n \uparrow\uparrow\uparrow n,1,2\}\)

\(g_{\Gamma_{\eta_0}}(n) \approx \{n,n \uparrow\uparrow\uparrow\uparrow n,1,2\}\)

\(g_{\Gamma_{\varphi(\omega,0)}}(n) \approx \{n,\{n,n,n+1\},1,2\}\)

\(g_{\Gamma_{\Gamma_0}}(n) \approx \{n,\{n,n,1,2\},1,2\}\)

\(g_{\Gamma_{\Gamma_{\Gamma_0}}}(n) \approx \{n,\{n,\{n,n,1,2\},1,2\},1,2\}\)

\(g_{\varphi(1,1,0)}(n) \approx \{n,n,2,2\}\)

\(g_{\varphi(1,2,0)}(n) \approx \{n,n,3,2\}\)

\(g_{\varphi(1,\omega,0)}(n) \approx \{n,n,n,2\}\)

\(g_{\varphi(1,\Gamma_0,0)}(n) \approx \{n,n,\{n,n,1,2\},2\}\)

\(g_{\varphi(1,\varphi(1,\Gamma_0,0),0)}(n) \approx \{n,n,\{n,n,\{n,n,1,2\},2\},2\}\)

\(g_{\varphi(2,0,0)}(n) \approx \{n,n,1,3\}\)

\(g_{\varphi(3,0,0)}(n) \approx \{n,n,1,4\}\)

\(g_{\varphi(\omega,0,0)}(n) \approx \{n,n,1,n+1\}\)

\(g_{\varphi(\Gamma_0,0,0)}(n) \approx \{n,n,1,\{n,n,1,2\}+1\}\)

\(g_{\varphi(1,0,0,0)}(n) \approx \{n,n,1,1,2\}\)

\(g_{\varphi(1,0,0,0,0)}(n) \approx \{n,n,1,1,1,2\}\)

\(g_{\varphi(1,0,0,0,0,0)}(n) \approx \{n,n,1,1,1,1,2\}\)

From \(\vartheta(\Omega^\omega)\) to \(\vartheta(\Omega^\Omega)\) Edit

\(g_{\vartheta(\Omega^\omega)}(n) \approx \{n,n+2 (1) 2\}\)

\(g_{\vartheta(\Omega^{\omega+1})}(n) \approx \{n,n+3 (1) 2\}\)

\(g_{\vartheta(\Omega^{\omega+m})}(n) \approx \{n,n+m+2 (1) 2\}\)

\(g_{\vartheta(\Omega^{\omega 2})}(n) \approx \{n,2n(1) 2\}\)

\(g_{\vartheta(\Omega^{\omega 3})}(n) \approx \{n,3n(1) 2\}\)

\(g_{\vartheta(\Omega^{\omega m})}(n) \approx \{n,m \times n(1) 2\}\)

\(g_{\vartheta(\Omega^{\omega^2})}(n) \approx \{n,n^2(1) 2\}\)

\(g_{\vartheta(\Omega^{\omega^\omega})}(n) \approx \{n,n^n(1) 2\}\)

\(g_{\vartheta(\Omega^{\varepsilon_0})}(n) \approx \{n,n\uparrow\uparrow n(1) 2\}\)

\(g_{\vartheta(\Omega^{\Gamma_0})}(n) \approx \{n,\{n,n,1,2\}(1)2\}\)

From \(\vartheta(\Omega^\Omega)\) to \(\vartheta(\Omega^{\Omega^\Omega})\) Edit

\(g_{\vartheta(\Omega^{\Omega})}(n) \approx \{n,n,2(1)2\}\)

\(g_{\vartheta(\Omega^{\Omega}+1)}(n) \approx \{n,n,3(1)2\}\)

\(g_{\vartheta(\Omega^{\Omega}+m)}(n) \approx \{n,n,m+2(1)2\}\)

\(g_{\vartheta(\Omega^{\Omega}+\omega)}(n) \approx \{n,n,n+2(1)2\}\)

\(g_{\vartheta(\Omega^{\Omega}+\Omega)}(n) \approx \{n,n,1,2(1)2\}\)

\(g_{\vartheta(\Omega^{\Omega}+\Omega \omega)}(n) \approx \{n,n,n,n(1)2\}\)

\(g_{\vartheta(\Omega^{\Omega}+\Omega^m \omega)}(n) \approx \{n,m+3(1)3\}\)

\(g_{\vartheta(\Omega^{\Omega}+\Omega^\omega)}(n) \approx \{n,n+3(1)3\}\)

\(g_{\vartheta(\Omega^{\Omega} 2)}(n) \approx \{n,n,2(1)3\}\)

\(g_{\vartheta(\Omega^{\Omega} 2+\Omega^\omega)}(n) \approx \{n,n+3(1)4\}\)

\(g_{\vartheta(\Omega^{\Omega} m+\Omega^\omega)}(n) \approx \{n,n+3(1)m+1\}\)

\(g_{\vartheta(\Omega^{\Omega} \omega)}(n) \approx \{n,n+3(1)n\}\)

\(g_{\vartheta(\Omega^{\Omega + 1})}(n) \approx \{n,n+1(1)1,2\}\)

\(g_{\vartheta(\Omega^{\Omega + m})}(n) \approx \{n,n+1(1)\underbrace{1,1...1,1}_{m},2\}\)

\(g_{\vartheta(\Omega^{\Omega 2})}(n) \approx \{n,n+1(1)(1)2\}\)

\(g_{\vartheta(\Omega^{\Omega 2 + m})}(n) \approx \{n,n+1(1)(1)\underbrace{1,1...1,1}_{m},2\}\)

\(g_{\vartheta(\Omega^{\Omega 3})}(n) \approx \{n,n+1(1)(1)(1)2\}\)

\(g_{\vartheta(\Omega^{\Omega m})}(n) \approx \{n,n+1\underbrace{(1)...(1)}_{m}2\}\)

\(g_{\vartheta(\Omega^{\Omega \omega})}(n) \approx \{n,n+1(2)2\}\)

\(g_{\vartheta(\Omega^{\Omega^2})}(n) \approx \{n,n,2(2)2\}\)

\(g_{\vartheta(\Omega^{\Omega^2}+\Omega^{\Omega \omega})}(n) \approx \{n,n+1(2)3\}\)

\(g_{\vartheta(\Omega^{\Omega^2}\omega)}(n) \approx \{n,n+1(2)n\}\)

\(g_{\vartheta(\Omega^{\Omega^2 + 1})}(n) \approx \{n,n(2)1,2\}\)

\(g_{\vartheta(\Omega^{\Omega^2 + \omega})}(n) \approx \{n,n(2)(1)2\}\)

\(g_{\vartheta(\Omega^{\Omega^2 + \Omega\omega})}(n) \approx \{n,n(2)(2)2\}\)

\(g_{\vartheta(\Omega^{\Omega^2\omega})}(n) \approx \{n,n+1(3)2\}\)

\(g_{\vartheta(\Omega^{\Omega^3})}(n) \approx \{n,n,2(3)2\}\)

\(g_{\vartheta(\Omega^{\Omega^{n-1}\omega})}(n) \approx \{n,n+1(n)2\}\)

\(g_{\vartheta(\Omega^{\Omega^\omega})}(n) \approx \{n,n+1(0,1)2\}\)

\(g_{\vartheta(\Omega^{\Omega^{\vartheta(\Omega^{\Omega^\omega})}})}(n) \approx \{n,\{n,n+1(0,1)2\}(0,1)2\}\)

From \(\vartheta(\Omega^{\Omega^\Omega})\) to \(\vartheta(\varepsilon_{\Omega + 1})\) Edit

\(g_{\vartheta(\Omega^{\Omega^\Omega})}(n) \approx \{n,n,2(0,1)2\}\)

\(g_{\vartheta(\Omega^{\Omega^\Omega + \Omega^\omega})}(n) \approx \{n,n(0,1)1(0,1)2\}\)

\(g_{\vartheta(\Omega^{\Omega^\Omega\omega})}(n) \approx \{n,n(1,1)2\}\)

\(g_{\vartheta(\Omega^{\Omega^{\Omega + \omega}})}(n) \approx \{n,n(0,2)2\}\)

\(g_{\vartheta(\Omega^{\Omega^{\Omega \omega}})}(n) \approx \{n,n(0,0,1)2\}\)

\(g_{\vartheta(\Omega^{\Omega^{\Omega^\omega}})}(n) \approx \{n,n((1)1)2\}\)

\(g_{\vartheta(\Omega^{\Omega^{\Omega^\Omega}})}(n) \approx \{n,n,2((1)1)2\}\)

\(g_{\vartheta(\Omega^{\Omega^{\Omega^{\Omega^\omega}}})}(n) \approx \{n,n((0,1)1)2\}\)

\(g_{\vartheta(\varepsilon_{\Omega+1})}(n) \approx X \uparrow\uparrow X\ \&\ n\)

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