緩増加関数は順序数\(\alpha\)に対し関数\(g_\alpha: \mathbb{N} \rightarrow \mathbb{N}\)を定義する一連の階層である。名前が示す通り、急増加関数やハーディー階層よりもより遅む成長する。
関数は次のように定義される:
- \(g_0(n) = 0\)
- \(g_{\alpha+1}(n) = g_\alpha(n)+1\)
- \(g_\alpha(n) = g_{\alpha[n]}(n)\) (\(\alpha\)は極限順序数)
\(\alpha[n]\)は順序数\(\alpha\)の基本列の\(n\)番目の項を表す。\(\alpha[n]\)の定義はあいまいであるため、緩増加関数と違ったものを作ることもできる。例えば、ワイナー階層、は急増加関数の項で説明されている。
小さな順序数に対しては、SGHはFGHと近接している。\(g_{\epsilon_0}(n)\)は\(f_3(n)\)ほどで、 \(f_{\varepsilon_0}(n)\)にSCGが追い付くのはバッハマン・ハワード順序数の所である。他のものと異なり、SGHは基本列の定義の変化に敏感である。一つのバージョンではSCGはFGHに\(\psi_0(\Omega_\omega)\)の所で追いつく。
巨大数論者にとって、SGHはFGH程には使いにくい。たが、これは順序数階層の中で最も遅く成長し、このことよりSGHは関数の成長率を階層化するのに最適かもしれない。面白いことに、BEAFではそれが自然に発生する。もしパイロットが\(\alpha\)の位置にあれば、プライムの変数の数は\(g_{\alpha}(p)\)である。
関数
\(\Gamma_0\)まで
\(g_0(n) = 0\)
\(g_1(n) = 1\)
\(g_2(n) = 2\)
\(g_m(n) = m\)
\(g_\omega(n) = n\)
\(g_{\omega^{\omega}}(n) = n^n\)
\(g_{\omega^{\omega^{\omega}}}(n) = n^{n^{n}}\)
\(g_{\varepsilon_0}(n) = n \uparrow\uparrow n\) (Ε₀を参照)
\(g_{\varepsilon_1}(n) \approx n \uparrow\uparrow (2n)\)
\(g_{\varepsilon_2}(n) \approx n \uparrow\uparrow (3n)\)
\(g_{\varepsilon_{\omega}}(n) \approx n \uparrow\uparrow (n^2)\)
\(g_{\varepsilon_{\omega^2}}(n) \approx n \uparrow\uparrow (n^3)\)
\(g_{\varepsilon_{\omega^3}}(n) \approx n \uparrow\uparrow (n^4)\)
\(g_{\varepsilon_{\omega^{\omega}}}(n) \approx n \uparrow\uparrow (n^n)\)
\(g_{\varepsilon_{\varepsilon_0}}(n) \approx n \uparrow\uparrow (n \uparrow\uparrow n)\)
\(g_{\zeta_0}(n) \approx n \uparrow\uparrow\uparrow n\)
\(g_{\varepsilon_{\zeta_0+1}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow n\)
\(g_{\varepsilon_{\zeta_0+2}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (2n)\)
\(g_{\varepsilon_{\zeta_0 2}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n) \approx n \uparrow\uparrow\uparrow (n+1)\)
\(g_{\varepsilon_{\zeta_0 3}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (2(n \uparrow\uparrow\uparrow n))\)
\(g_{\varepsilon_{\zeta_0 4}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (3(n \uparrow\uparrow\uparrow n))\)
\(g_{\varepsilon_{\zeta_0 \omega}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n(n \uparrow\uparrow\uparrow n))\)
\(g_{\varepsilon_{\zeta_0^2}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ({(n \uparrow\uparrow\uparrow n)}^2)\)
\(g_{\varepsilon_{\zeta_0^{\zeta_0}}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ({(n \uparrow\uparrow\uparrow n)}^{n \uparrow\uparrow\uparrow n})\)
\(g_{\varepsilon_{\varepsilon_{\zeta_0+1}}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow n)\)
\(g_{\varepsilon_{\varepsilon_{\zeta_0 2}}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n)) \approx n \uparrow\uparrow\uparrow (n+2)\)
\(g_{\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0+1}}}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow n))\)
\(g_{\zeta_1}(n) \approx n \uparrow\uparrow\uparrow 2n\)
\(g_{\zeta_2}(n) \approx n \uparrow\uparrow\uparrow 3n\)
\(g_{\zeta_\omega}(n) \approx n \uparrow\uparrow\uparrow n^2\)
\(g_{\zeta_{\omega^\omega}}(n) \approx n \uparrow\uparrow\uparrow n^n\)
\(g_{\zeta_{\varepsilon_0}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow n)\)
\(g_{\zeta_{\varepsilon_{\varepsilon_0}}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow (n \uparrow\uparrow n))\)
\(g_{\zeta_{\zeta_0}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow\uparrow n)\)
\(g_{\zeta_{\zeta_{\zeta_0}}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow\uparrow (n \uparrow\uparrow\uparrow n\)))
\(g_{\eta_0}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\)
\(g_{\varepsilon_{\eta_0+1}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow n\)
\(g_{\varepsilon_{\eta_0+2}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow 2n\)
\(g_{\varepsilon_{\eta_0+\omega}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow n^2\)
\(g_{\varepsilon_{\eta_0+\omega^\omega}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow n^n\)
\(g_{\varepsilon_{\eta_0+\varepsilon_0}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow n)\)
\(g_{\varepsilon_{\eta_0+\varepsilon_{\varepsilon_0}}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow n \uparrow\uparrow n)\)
\(g_{\varepsilon_{\eta_0+\zeta_0}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n)\)
\(g_{\varepsilon_{\eta_0+\zeta_1}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow 2n)\)
\(g_{\varepsilon_{\eta_0+\zeta_\omega}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n^2)\)
\(g_{\varepsilon_{\eta_0+\zeta_{\omega^\omega}}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n^n)\)
\(g_{\varepsilon_{\eta_0+\zeta_{\zeta_0}}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n \uparrow\uparrow\uparrow n)\)
\(g_{\varepsilon_{\eta_0 2}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow\uparrow n)\)
\(g_{\varepsilon_{\eta_0 3}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow 2(n \uparrow\uparrow\uparrow\uparrow n)\)
\(g_{\varepsilon_{\eta_0 \omega}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow n-1(n \uparrow\uparrow\uparrow\uparrow n)\)
\(g_{\zeta_{\eta_0+1}}(n) \approx (n \uparrow\uparrow\uparrow\uparrow n) \uparrow\uparrow\uparrow n\)
\(g_{\varphi(4,0)}(n) \approx n \uparrow\uparrow\uparrow\uparrow\uparrow n\)
\(g_{\varphi(5,0)}(n) \approx n \uparrow\uparrow\uparrow\uparrow\uparrow\uparrow n\)
\(g_{\varphi(\omega,0)}(n) \approx \{n,n,n+1\}\)
\(g_{\varphi(\omega^\omega,0)}(n) \approx \{n,n,n^n+1\}\)
\(g_{\varphi(\varepsilon_0,0)}(n) \approx \{n,n,n \uparrow\uparrow n+1\}\)
\(g_{\varphi(\zeta_0,0)}(n) \approx \{n,n,n \uparrow\uparrow\uparrow n+1\}\)
\(g_{\varphi(\eta_0,0)}(n) \approx \{n,n,n \uparrow\uparrow\uparrow\uparrow n+1\}\)
\(g_{\varphi(\varphi(\omega,0),0)}(n) \approx \{n,n,\{n,n,n+1\}+1\}\)
\(g_{\varphi(\varphi(\varphi(\omega,0),0),0)}(n) \approx \{n,n,\{n,n,\{n,n,n+1\}+1\}+1\}\)
\(\Gamma_0\)から\(\vartheta(\Omega^\omega)\)
\(g_{\Gamma_0}(n) \approx \{n,n,1,2\}\)
\(g_{\varphi(\Gamma_0,1)}(n) \approx \{n,n+1,1,2\}\)
\(g_{\varphi(\varphi(\Gamma_0,1),0)}(n) \approx \{n,n+2,1,2\}\)
\(g_{\Gamma_1}(n) \approx \{n,2n,1,2\}\)
\(g_{\Gamma_2}(n) \approx \{n,3n,1,2\}\)
\(g_{\Gamma_\omega}(n) \approx \{n,(n+1)n,1,2\}\)
\(g_{\Gamma_{\omega^2}}(n) \approx \{n,(n^2+1)n,1,2\}\)
\(g_{\Gamma_{\omega^\omega}}(n) \approx \{n,(n^{n-1}+1)n,1,2\}\)
\(g_{\Gamma_{\omega^{\omega^\omega}}}(n) \approx \{n,(n^{n^n-1}+1)n,1,2\}\)
\(g_{\Gamma_{\varepsilon_0}}(n) \approx \{n,n \uparrow\uparrow n,1,2\}\)
\(g_{\Gamma_{\zeta_0}}(n) \approx \{n,n \uparrow\uparrow\uparrow n,1,2\}\)
\(g_{\Gamma_{\eta_0}}(n) \approx \{n,n \uparrow\uparrow\uparrow\uparrow n,1,2\}\)
\(g_{\Gamma_{\varphi(\omega,0)}}(n) \approx \{n,\{n,n,n+1\},1,2\}\)
\(g_{\Gamma_{\Gamma_0}}(n) \approx \{n,\{n,n,1,2\},1,2\}\)
\(g_{\Gamma_{\Gamma_{\Gamma_0}}}(n) \approx \{n,\{n,\{n,n,1,2\},1,2\},1,2\}\)
\(g_{\varphi(1,1,0)}(n) \approx \{n,n,2,2\}\)
\(g_{\varphi(1,2,0)}(n) \approx \{n,n,3,2\}\)
\(g_{\varphi(1,\omega,0)}(n) \approx \{n,n,n,2\}\)
\(g_{\varphi(1,\Gamma_0,0)}(n) \approx \{n,n,\{n,n,1,2\},2\}\)
\(g_{\varphi(1,\varphi(1,\Gamma_0,0),0)}(n) \approx \{n,n,\{n,n,\{n,n,1,2\},2\},2\}\)
\(g_{\varphi(2,0,0)}(n) \approx \{n,n,1,3\}\)
\(g_{\varphi(3,0,0)}(n) \approx \{n,n,1,4\}\)
\(g_{\varphi(\omega,0,0)}(n) \approx \{n,n,1,n+1\}\)
\(g_{\varphi(\Gamma_0,0,0)}(n) \approx \{n,n,1,\{n,n,1,2\}+1\}\)
\(g_{\varphi(1,0,0,0)}(n) \approx \{n,n,1,1,2\}\)
\(g_{\varphi(1,0,0,0,0)}(n) \approx \{n,n,1,1,1,2\}\)
\(g_{\varphi(1,0,0,0,0,0)}(n) \approx \{n,n,1,1,1,1,2\}\)
From \(\vartheta(\Omega^\omega)\) to \(\vartheta(\Omega^\Omega)\)
\(g_{\vartheta(\Omega^\omega)}(n) \approx \{n,n+2 (1) 2\}\)
\(g_{\vartheta(\Omega^{\omega+1})}(n) \approx \{n,n+3 (1) 2\}\)
\(g_{\vartheta(\Omega^{\omega+m})}(n) \approx \{n,n+m+2 (1) 2\}\)
\(g_{\vartheta(\Omega^{\omega 2})}(n) \approx \{n,2n(1) 2\}\)
\(g_{\vartheta(\Omega^{\omega 3})}(n) \approx \{n,3n(1) 2\}\)
\(g_{\vartheta(\Omega^{\omega m})}(n) \approx \{n,m*n(1) 2\}\)
\(g_{\vartheta(\Omega^{\omega^2})}(n) \approx \{n,n^2(1) 2\}\)
\(g_{\vartheta(\Omega^{\omega^\omega})}(n) \approx \{n,n^n(1) 2\}\)
\(g_{\vartheta(\Omega^{\varepsilon_0})}(n) \approx \{n,n\uparrow\uparrow n(1) 2\}\)
\(g_{\vartheta(\Omega^{\Gamma_0})}(n) \approx \{n,\{n,n,1,2\}(1)2\}\)
\(\vartheta(\Omega^\Omega)\)から\(\vartheta(\Omega^{\Omega^\Omega})\)
\(g_{\vartheta(\Omega^{\Omega})}(n) \approx \{n,n,2(1)2\}\)
\(g_{\vartheta(\Omega^{\Omega}+1)}(n) \approx \{n,n,3(1)2\}\)
\(g_{\vartheta(\Omega^{\Omega}+m)}(n) \approx \{n,n,m+2(1)2\}\)
\(g_{\vartheta(\Omega^{\Omega}+\omega)}(n) \approx \{n,n,n+2(1)2\}\)
\(g_{\vartheta(\Omega^{\Omega}+\Omega)}(n) \approx \{n,n,1,2(1)2\}\)
\(g_{\vartheta(\Omega^{\Omega}+\Omega \omega)}(n) \approx \{n,n,n,n(1)2\}\)
\(g_{\vartheta(\Omega^{\Omega}+\Omega^m \omega)}(n) \approx \{n,m+3(1)3\}\)
\(g_{\vartheta(\Omega^{\Omega}+\Omega^\omega)}(n) \approx \{n,n+3(1)3\}\)
\(g_{\vartheta(\Omega^{\Omega} 2)}(n) \approx \{n,n,2(1)3\}\)
\(g_{\vartheta(\Omega^{\Omega} 2+\Omega^\omega)}(n) \approx \{n,n+3(1)4\}\)
\(g_{\vartheta(\Omega^{\Omega} m+\Omega^\omega)}(n) \approx \{n,n+3(1)m+1\}\)
\(g_{\vartheta(\Omega^{\Omega} \omega)}(n) \approx \{n,n+3(1)n\}\)
\(g_{\vartheta(\Omega^{\Omega + 1})}(n) \approx \{n,n+1(1)1,2\}\)
\(g_{\vartheta(\Omega^{\Omega + m})}(n) \approx \{n,n+1(1)\underbrace{1,1...1,1}_{m},2\}\)
\(g_{\vartheta(\Omega^{\Omega 2})}(n) \approx \{n,n+1(1)(1)2\}\)
\(g_{\vartheta(\Omega^{\Omega 2 + m})}(n) \approx \{n,n+1(1)(1)\underbrace{1,1...1,1}_{m},2\}\)
\(g_{\vartheta(\Omega^{\Omega 3})}(n) \approx \{n,n+1(1)(1)(1)2\}\)
\(g_{\vartheta(\Omega^{\Omega m})}(n) \approx \{n,n+1\underbrace{(1)...(1)}_{m}2\}\)
\(g_{\vartheta(\Omega^{\Omega \omega})}(n) \approx \{n,n+1(2)2\}\)
\(g_{\vartheta(\Omega^{\Omega^2})}(n) \approx \{n,n,2(2)2\}\)
\(g_{\vartheta(\Omega^{\Omega^2}+\Omega^{\Omega \omega})}(n) \approx \{n,n+1(2)3\}\)
\(g_{\vartheta(\Omega^{\Omega^2}\omega)}(n) \approx \{n,n+1(2)n\}\)
\(g_{\vartheta(\Omega^{\Omega^2 + 1})}(n) \approx \{n,n(2)1,2\}\)
\(g_{\vartheta(\Omega^{\Omega^2 + \omega})}(n) \approx \{n,n(2)1(1)2\}\)
\(g_{\vartheta(\Omega^{\Omega^2 + \Omega\omega})}(n) \approx \{n,n(2)1(2)2\}\)
\(g_{\vartheta(\Omega^{\Omega^2\omega})}(n) \approx \{n,n+1(3)2\}\)
\(g_{\vartheta(\Omega^{\Omega^3})}(n) \approx \{n,n,2(3)2\}\)
\(g_{\vartheta(\Omega^{\Omega^{n-1}\omega})}(n) \approx \{n,n+1(n)2\}\)
\(g_{\vartheta(\Omega^{\Omega^\omega})}(n) \approx \{n,n+1(0,1)2\}\)
\(g_{\vartheta(\Omega^{\Omega^{\vartheta(\Omega^{\Omega^\omega})}})}(n) \approx \{n,\{n,n+1(0,1)2\}(0,1)2\}\)
\(\vartheta(\Omega^{\Omega^\Omega})\)から\(\vartheta(\varepsilon_{\Omega + 1})\)
\(g_{\vartheta(\Omega^{\Omega^\Omega})}(n) \approx \{n,n,2(0,1)2\}\)
\(g_{\vartheta(\Omega^{\Omega^\Omega + \Omega^\omega})}(n) \approx \{n,n(0,1)1(0,1)2\}\)
\(g_{\vartheta(\Omega^{\Omega^\Omega\omega})}(n) \approx \{n,n(1,1)2\}\)
\(g_{\vartheta(\Omega^{\Omega^{\Omega + \omega}})}(n) \approx \{n,n(0,2)2\}\)
\(g_{\vartheta(\Omega^{\Omega^{\Omega \omega}})}(n) \approx \{n,n(0,0,1)2\}\)
\(g_{\vartheta(\Omega^{\Omega^{\Omega^\omega}})}(n) \approx \{n,n((1)1)2\}\)
\(g_{\vartheta(\Omega^{\Omega^{\Omega^\Omega}})}(n) \approx \{n,n,2((1)1)2\}\)
\(g_{\vartheta(\Omega^{\Omega^{\Omega^{\Omega^\omega}}})}(n) \approx \{n,n((0,1)1)2\}\)
\(\vartheta(\varepsilon_{\Omega + 1})\)から\(\vartheta(\Omega_2)\)
\(g_{\vartheta(\varepsilon_{\Omega + 1})}(n) \approx \{n,n(X \uparrow\uparrow X)2\}\)
\(g_{\vartheta(\varepsilon_{\Omega 2})}(n) \approx \{n,n,2(X \uparrow\uparrow X)2\}\)
\(g_{\vartheta(\varepsilon_{\Omega 2}^\omega)}(n) \approx \{n,n(X \uparrow\uparrow X + 1)2\}\)
\(g_{\vartheta(\varepsilon_{\Omega 2}^{\Omega^\omega})}(n) \approx \{n,n(X \uparrow\uparrow X + X)2\}\)
\(g_{\vartheta(\varepsilon_{\Omega 2}^{\Omega^{\Omega^\omega}})}(n) \approx \{n,n(X \uparrow\uparrow X + X^X)2\}\)
\(g_{\vartheta(\varepsilon_{\Omega 2}^{\varepsilon_{\Omega + 1}})}(n) \approx \{n,n((X \uparrow\uparrow X)*2)2\}\)
\(g_{\vartheta(\varepsilon_{\Omega 2}^{\varepsilon_{\Omega 2}\omega})}(n) \approx \{n,n((X \uparrow\uparrow X)*X)2\}\)
\(g_{\vartheta(\varepsilon_{\Omega 2}^{\varepsilon_{\Omega 2}\varepsilon_{\Omega + 1}})}(n) \approx \{n,n((X \uparrow\uparrow X)^2)2\}\)
\(g_{\vartheta(\varepsilon_{\Omega 2}^{\varepsilon_{\Omega 2}^\omega})}(n) \approx \{n,n((X \uparrow\uparrow X)^X)2\}\)
\(g_{\vartheta(\varepsilon_{\Omega 2}^{\varepsilon_{\Omega 2}^{\varepsilon_{\Omega + 1}}})}(n) \approx \{n,n((X \uparrow\uparrow X)^{(X \uparrow\uparrow X)})2\}\)