Ε₀

\(\varepsilon_0\) (pronounced "epsilon-zero", "epsilon-null" or "epsilon-naught") is a small countable ordinal, defined as the first fixed point of the function \(\omega^\alpha \mapsto \alpha\). It can be defined in several ways:


 * The first ordinal for which Hardy hierarchy catches up the fast-growing one.
 * Informal visualization: \(\omega^{\omega^{\omega^{.^{.^.}}}}\)
 * The first solution to the equation \(\omega^\alpha = \alpha\).
 * \(\omega \uparrow\uparrow \omega\)
 * The proof-theoretic ordinal of Peano arithmetic and arithmetical comprehension.

The growth rates of finite forms of the 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)

\(f_{\varepsilon_0}(n)\) is comparable to the Goodstein function and Goucher's T(n) function.

At \(\varepsilon_0\), the Hardy hierarchy catches up to the fast-growing hierarchy.

Higher epsilon numbers and the Veblen hierarchy
The function \(\alpha \mapsto \varepsilon_\alpha\) enumerates the fixed points of the exponential map \(\alpha \mapsto \omega^\alpha\). Thus \(\varepsilon_1\) is the next fixed point of the exponential map, informally visualized as \(\omega \uparrow\uparrow (\omega 2)\), and in general \(\varepsilon_\alpha\) could be called \(\omega \uparrow\uparrow (\omega (1+\alpha))\).

The limit of the epsilon numbers is the first fixed point of \(\alpha \mapsto \varepsilon_\alpha\), visualized as \(\varepsilon_{\varepsilon_{\varepsilon_{._{._.}}}} = \omega \uparrow\uparrow (\omega + \omega \uparrow\uparrow (\omega + \ldots)) = \omega \uparrow\uparrow\uparrow \omega\). This number is called \(\zeta_0\) (zeta-zero), and \(\zeta_\alpha\) (\(\omega \uparrow\uparrow\uparrow (\omega + \alpha)\)) enumerates the fixed points of \(\alpha \mapsto \varepsilon_\alpha\).

Since we do not have an infinite number of Greek letters, we generalize this using a series of functions that form the Veblen hierarchy. Each function enumerates the fixed points of the previous one. Formally:


 * \(\phi_0(\alpha) = \omega^\alpha\)
 * \(\phi_\beta(\alpha)\) is the \(\alpha\)th fixed point of \(\phi_\gamma\) for all \(\gamma < \beta\)

The first ordinal inaccessible through the Veblen hierarchy is Gamma-zero.