\(\varepsilon_0\) (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 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.

External links Edit

See also Edit

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence
Theories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...