## FANDOM

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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:

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

## Higher epsilon numbers and the Veblen hierarchy Edit

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.

The limit of the epsilon numbers is the first fixed point of $$\alpha \mapsto \varepsilon_\alpha$$. This ordinal is called $$\zeta_0$$ (zeta-zero), and $$\zeta_\alpha$$ 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 this two-argument Veblen hierarchy is the Feferman–Schütte ordinal.