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Not to be confused with ε function.

\(\varepsilon_0\) (pronounced "epsilon-zero", "epsilon-null" or "epsilon-nought") is a small countable ordinal, defined as the first fixed point of the function \(\alpha \mapsto \omega^\alpha\)[1].

Properties

The ordinal \(\varepsilon_0\) has several properties:

  • Smallest ordinal not expressible in Cantor normal form 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 Peano arithmetic and ACA0 (arithmetical comprehension, a subsystem of second-order arithmetic).[2]
  • 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

Appearance in googology

Using the Wainer 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

  1. D. Madore, A Zoo of Ordinals (p.1). Accessed 2021-05-30.
  2. W. Pohlers, Proof theory: The first step into impredicativity, Springer, 2009.

See also

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

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