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\(\varepsilon_0\) (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)

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  • \(\varepsilon_0\) (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)
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  • \(\varepsilon_0\) (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: * Smallest ordinal not expressible in Cantor normal form using strictly smaller exponents. * The proof-theoretic ordinal of Peano arithmetic and ACA0 (arithmetical comprehension, a subsystem of second-order arithmetic). * Informal visualizations: \(\omega^{\omega^{\omega^{.^{.^.}}}}\) or \(\omega \uparrow\uparrow \omega\) or \(\omega \uparrow\uparrow\uparrow 2\) 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) \(f_{\varepsilon_0}(n)\) is comparable to the Goodstein function and Goucher's T(n) function.
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