Ε₀

\(\varepsilon_0\) (pronounced "epsilon-zero", "epsilon-null" or "epsilon-nought") is a small countable ordinal, defined as the first fixed point of the function \(\alpha \mapsto \omega^\alpha\).

Properties
The ordinal \(\varepsilon_0\) has several properties:
 * Smallest ordinal not expressible in using strictly smaller exponents. Equivalently, this is the least nonzero ordinal that is closed under addition and \(\lambda\alpha.\omega^\alpha\).
 * In fact, \(\varepsilon_0\) is also the least nonzero ordinal closed under only \(\lambda\alpha.\omega^\alpha\).
 * The proof-theoretic ordinal of and ACA0 (arithmetical comprehension, a subsystem of second-order arithmetic).
 * Informal visualizations: \(\omega^{\omega^{\omega^\cdots}}\) or \(\omega \uparrow\uparrow \omega\)
 * The second fixed point of \(x\mapsto2^x\).
 * \(\varphi(1,0)\) using the Veblen function
 * \(\psi_0(\Omega)\) using Buchholz's function
 * \(\psi(0)\) using Madore's function
 * \({}^{\omega}\omega\) using Rucker's Tetration

Appearance in googology
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)

\(f_{\varepsilon_0}(n)\) with respect to the Wainer hierarchy is comparable to the Goodstein function, Beklevmishev's worm function \(\textrm{Worm}(n)\), and its variants.

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 after \(\varepsilon_0\). 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 non-zero limit ordinals smaller than \(\zeta_0\) explained later:
 * If \(\alpha=\beta_1+\cdots+\beta_{k-1}+\beta_k\), where \(k\) is a natural number greater than \(1\) and \(\beta_1,\ldots,\beta_{k-1},\beta_k\) are additive principal ordinals greater than \(1\) satisfying \(\beta_1 \geq \cdots \geq \beta_{k-1} \geq \beta_k\), then \(\alpha[n]=\beta_1+\cdots+\beta_{k-1}+(\beta_k[n])\)
 * If \(\alpha=\omega^{\beta+1}\), then \(\alpha[n]=\omega^{\beta} \times n\)
 * If \(\alpha=\omega^{\beta}\) and \(\beta\) is a non-zero limit ordinal which is not an epsilon number, then \(\alpha[n]=\omega^{\beta[n]}\)
 * 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 non-zero limit ordinal, then \(\alpha[n]=\varepsilon_{\beta[n]}\)

The first fixed point of \(\alpha \mapsto \varepsilon_\alpha\) 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.