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 \(\alpha \mapsto \omega^\alpha\)D. Madore, A Zoo of Ordinals (p.1). Accessed 2021-05-30..
==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).W. Pohlers, Proof theory: The first step into impredicativity, Springer, 2009.
* Informal visualizations: \(^\omega\omega\), \(\omega^{\omega^{\omega^\cdots}}\) 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
==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.
== Sources ==
== See also ==
ja:ε₀
Category:Transfinite ordinals