Ε₀

The \(\varepsilon_0\) (pronounced "epsilon-zero", "epsilon-null" or "epsilon-naught") is the first ordinal satisfying the equation \(\omega^{\alpha} = \alpha\). The growth rates of finite forms of that ordinal in different hierarchies are shown below:

\(f_{\varepsilon_0}(n) \approx X \uparrow\uparrow X \&\ n\) (fast-growing hierarchy)

\(H_{\varepsilon_0}(n) \approx X \uparrow\uparrow X \&\ n\) (Hardy hierarchy)

\(g_{\varepsilon_0}(n) = n \uparrow\uparrow n\) (slow-growing hierarchy)

In the fast-growing and Hardy hierarchies it has applications of functions with comparable growth rates: Goodstein function and Goucher's T(n) function.