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The small Veblen ordinal is the limit of the following sequence of ordinals \(\varphi(1, 0),\varphi(1, 0, 0),\varphi(1, 0, 0, 0),\varphi(1, 0, 0, 0, 0),\ldots\) where \(\varphi\) is the Veblen hierarchy as extended to arbitrary finite numbers of arguments. Using Buchholz's function it can be expressed as \(\psi_0(\psi_1(\psi_1(\psi_1(1))))\), and using Weiermann's \(\vartheta\) function, it can be expressed as \(\vartheta(\Omega^\omega)\). Demonstrating the power of the Veblen hierarchy, this ordinal has been remarked as much greater than \(\varphi(\omega,0)\).[1]

For the growth rate of Harvey Friedman's tree(n) function (for unlabeled trees), a strict inequality \(\textrm{tree}(1.0001n+2) > f_{\vartheta(\Omega^\omega)}(n)\) for \(n \geq 3\) was proved.[2]

This ordinal is the proof-theoretic ordinal of Kripke-Platek set theory with \(\Delta_0\)-induction along \(\mathbb{N}\) and foundation restricted to \(\Pi_2\) formulas.[3]

In the fast-growing hierarchy, Bird insisted that using Bird's array notation \(f_{\vartheta(\Omega^{\omega})}(n)\) is approximately {n,n [1 [1 ¬ 1,2] 2] 2} in the Nested Hyper-Nested Array Notation,[4] and {n,n [1 [1 \ 1,2 ~ 2] 2] 2} in the Hierarchial Hyper-Nested Array Notation.[5]

In the slow-growing hierarchy, \(g_{\vartheta(\Omega^{\omega})}(n)\) is approximately {n,n (1) 2} in BEAF.


  1. I. Lepper, G. Moser, Why ordinals are good for you (p.2). Accessed 2021-06-16.
  2. T. Kihara, Lower bounds for tree(4) and tree(5), とりマセ Σ^0_2, 05/2020.
  3. Michael Rathjen, "Fragments of Kripke–Platek Set Theory with Infinity" (a survey without a proof or a reference to the first source)
  4. Bird, Chris. Beyond Bird’s Nested Arrays II.
  5. Bird, Chris. Beyond Bird’s Nested Arrays IV.

See also[]

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
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 one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak 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 · Arai'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: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)