# Slow-growing hierarchy

The slow-growing hierarchy is a family of functions $$(g_\alpha:\mathbb N\rightarrow\mathbb N)_{\alpha<\mu}$$ where $$\mu$$ is a large countable ordinal such that a fundamental sequence is assigned for each limit ordinal less than $$\mu$$.

The slow-growing hierarchy 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,

where $$\alpha[n]$$ denotes the $$n$$th element of the fundamental sequence assigned to the limit ordinal $$\alpha$$.

Every nonzero ordinal $$\alpha<\varepsilon_0=\min\{\beta|\beta=\omega^\beta\}$$ can be represented in a unique Cantor normal form $$\alpha=\omega^{\beta_{1}}+ \omega^{\beta_{2}}+\cdots+\omega^{\beta_{k-1}}+\omega^{\beta_{k}}$$ where $$\alpha>\beta_1\geq\beta_2\geq\cdots\geq\beta_{k-1}\geq\beta_k$$.

If $$\beta_k>0$$ then $$\alpha$$ is a limit and we can assign to it a fundamental sequence as follows

$$\alpha[n]=\omega^{\beta_{1}}+ \omega^{\beta_{2}}+\cdots+\omega^{\beta_{k-1}}+\left\{\begin{array}{lcr} \omega^\gamma n \text{ if } \beta_k=\gamma+1\\ \omega^{\beta_k[n]} \text{ if } \beta_k \text{ is a limit.}\\ \end{array}\right.$$

If $$\alpha=\varepsilon_0$$ then $$\alpha[0]=0$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$.

Using this system of fundamental sequences we can define the slow-growing hierarchy up to $$\varepsilon_0$$ and we have $$g_{\varepsilon_0}(n) = n \uparrow\uparrow n$$

There are much stronger systems of fundamental sequences you can see on the following pages:

## Values

$$g_0(n) = 0$$

$$g_1(n) = 1$$

$$g_2(n) = 2$$

$$g_m(n) = m$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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