\(\varepsilon_0\) (pronounced "epsilon-zero", "epsilon-null" or "epsilon-nought") is a small countable ordinal, defined as the first fixed point of the function \(\alpha \mapsto \omega^\alpha\). It can also be equivalently defined in several other ways: Using the Wainer hierarchy: * \(f_{\varepsilon_0}(n) \approx X \uparrow\uparrow X\ \&\ n\) (fast-growing hierarchy) * \(H_{\varepsilon_0}(n) \approx X \uparrow\uparrow X\ \&\ n\) (Hardy hierarchy) * \(g_{\varepsilon_0}(n) = n \uparrow\uparrow n = n \uparrow\uparrow\uparrow 2\) (slow-growing hierarchy)