Difference between revisions of "Slow-growing hierarchy"

From Cantor's Attic
Jump to: navigation, search
(Values)
Line 5: Line 5:
 
\(g_\alpha(n)=g_{\alpha[n]}(n)\) if and only if \(\alpha\) is a limit ordinal.
 
\(g_\alpha(n)=g_{\alpha[n]}(n)\) if and only if \(\alpha\) is a limit ordinal.
  
 +
The slow-growing hierarchy is not very useful to googologists, but it can reveal the direct relationship between \(\omega\) and other ordinals.
  
 
== Values ==
 
== Values ==

Revision as of 03:50, 15 April 2017

The slow-growing hierarchy is a hierarchy like the Fast-growing hierarchy, though this group of functions are slow-growing. It is defined as follows:

\(g_0(n)=0 \\ g_{\alpha+1}(n)=g_\alpha(n)+1\)

\(g_\alpha(n)=g_{\alpha[n]}(n)\) if and only if \(\alpha\) is a limit ordinal.

The slow-growing hierarchy is not very useful to googologists, but it can reveal the direct relationship between \(\omega\) and other ordinals.

Values

\(g_0(n) = 0\)

\(g_1(n) = 1\)

\(g_2(n) = 2\)

\(g_m(n) = m\)

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

\(g_{\omega+1}(n) = n+1 = f_0(n)\)

\(g_{\omega2}(n) = f_1(n)\)

\(g_{\omega^{\omega}}(n) = n^n \approx f_2(n)\)

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

\(g_{\varepsilon_0}(n) = n \uparrow\uparrow n \approx f_3(n)\)

\(g_{\varepsilon_1}(n) = E(n)(n \uparrow\uparrow n+1)\#n \approx n \uparrow\uparrow (2n)\)(see Hyper-E notation)

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

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

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

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

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

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

\(g_{\zeta_0}(n) \approx n \uparrow\uparrow\uparrow n \approx f_4(n)\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(g_{\eta_0}(n) \approx n \uparrow\uparrow\uparrow\uparrow n \approx f_5(n)\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(g_{\varphi(4,0)}(n) \approx n \uparrow\uparrow\uparrow\uparrow\uparrow n \approx f_6(n)\)

\(g_{\varphi(5,0)}(n) \approx n \uparrow\uparrow\uparrow\uparrow\uparrow\uparrow n \approx f_7(n)\)

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

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

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

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

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

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

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

\(g_{\Gamma_0}(n) \approx \{n,n,1,2\} < f_{\omega+1}(n)\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(g_{\varphi(1,2,0)}(n) \approx \{n,n,3,2\} < f_{\omega+3}(n)\)

\(g_{\varphi(1,\omega,0)}(n) \approx \{n,n,n,2\} < f_{\omega2}(n)\)

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

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

\(g_{\varphi(2,0,0)}(n) \approx \{n,n,1,3\} < f_{\omega2+1}(n)\)

\(g_{\varphi(3,0,0)}(n) \approx \{n,n,1,4\} < f_{\omega3+1}(n)\)

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

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

\(g_{\varphi(1,0,0,0)}(n) \approx \{n,n,1,1,2\} < f_{\omega^2+1}(n)\)

\(g_{\varphi(1,0,0,0,0)}(n) \approx \{n,n,1,1,1,2\} < f_{\omega^3+1}(n)\)

\(g_{\varphi(1,0,0,0,0,0)}(n) \approx \{n,n,1,1,1,1,2\} < f_{\omega^4+1}(n)\)

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

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

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

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

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

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

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

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

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

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

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

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

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

\(g_{\vartheta(\Omega^{\Omega}+\Omega^\omega)}(n) \approx \{n,n(1)3\} < f_{\omega^\omega2}(n)\)

\(g_{\vartheta(\Omega^{\Omega} 2)}(n) \approx \{n,n,2(1)3\} < f_{\omega^\omega2+1}(n)\)

\(g_{\vartheta(\Omega^{\Omega} 2+\Omega^\omega)}(n) \approx \{n,n(1)4\} < f_{\omega^\omega3}(n)\)

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

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

\(g_{\vartheta(\Omega^{\Omega 2})}(n) \approx \{n,n(1)(1)2\} < f_{\omega^{\omega2}}(n)\)

\(g_{\vartheta(\Omega^{\Omega 3})}(n) \approx \{n,n(1)(1)(1)2\} < f_{\omega^{\omega3}}(n)\)

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

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

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

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

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

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

\(g_{\vartheta(\Omega^{\Omega^2 + \Omega\omega})}(n) \approx \{n,n(2)(2)2\} < f_{\omega^{\omega^22}}(n)\)

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

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

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

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

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

\(g_{\vartheta(\Omega^{\Omega^\Omega + \Omega^\omega})}(n) \approx \{n,n(0,1)1(0,1)2\} < f_{\omega^{\omega^\omega2}}(n)\)

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

\(g_{\vartheta(\Omega^{\Omega^{\Omega + \omega}})}(n) \approx \{n,n(0,2)2\} < f_{\omega^{\omega^{\omega2}}}(n)\)

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

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

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

\(g_{\vartheta(\Omega^{\Omega^{\Omega^{\Omega^\omega}}})}(n) \approx \{n,n((0,1)1)2\} < f_{^5 \omega}(n)\)

\(g_{\vartheta(\varepsilon_{\Omega+1})}(n) \approx X^X \&\ n < f_{\varepsilon_0}(n)\)