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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 $$\omega \uparrow\uparrow\uparrow 2$$
• The second fixed point of $$x\mapsto2^x$$.
• $$\psi_0(\Omega)$$ using Buchholz notation
• $$\psi(0)$$ using Madore’s notation

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. Formally:

• $$\varepsilon_0=\text{min}\{\alpha|\alpha=\omega^\alpha\}=\text{sup}\{0,1,\omega, \omega^\omega, \omega^{\omega^\omega},...\}$$
• $$\varepsilon_{\alpha+1}=\text{min}\{\beta|\beta=\omega^\beta\wedge\beta>\varepsilon_\alpha\}=\text{sup}\{\varepsilon_\alpha+1,\omega^{\varepsilon_\alpha+1}, \omega^{\omega^{\varepsilon_\alpha+1}},...\}$$
• $$\varepsilon_{\alpha}=\text{sup}\{\varepsilon_{\beta}|\beta<\alpha\}$$ if $$\alpha$$ is a limit ordinal.

This definition gives the following fundamental sequences for epsilon numbers:

• if $$\alpha=\varepsilon_0$$ then $$\alpha[0]=0$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$
• if $$\alpha=\varepsilon_{\beta+1}$$ then $$\alpha[0]=\varepsilon_\beta+1$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$
• if $$\alpha=\varepsilon_{\beta}$$ and $$\beta$$ is a limit ordinal then $$\alpha[n]=\varepsilon_{\beta[n]}$$

The limit of the epsilon numbers is the first fixed point of $$\alpha \mapsto \varepsilon_\alpha$$. This ordinal is called $$\zeta_0$$ (zeta-zero) or Cantor's ordinal, 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:

• $$\varphi_0(\alpha) = \omega^\alpha$$
• $$\varphi_\beta(\alpha)$$ is the $$(1+\alpha)$$th fixed point of $$\varphi_\gamma$$ for all $$\gamma < \beta$$

The first ordinal inaccessible through this two-argument Veblen hierarchy is the Feferman–Schütte ordinal.

## Edit

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