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The Bachmann-Howard ordinal is a large countable ordinal, significant for being the proof-theoretic ordinal of Kripke-Platek set theory with the axiom of infinity[1][2][3][footnote 1]. It is the supremum \(\vartheta(\varepsilon_{\Omega+1})\) of \(\vartheta(\alpha)\) for all \(\alpha < \varepsilon_{\Omega+1}\) with respect to Weiermann's \(\vartheta\) and is presented as \(\psi_0(\Omega_2) = \psi_0(\psi_2(0))\) with respect to Buchholz's \(\psi\). The Bachmann-Howard ordinal has also been denoted as \(\eta_0\)[4], however this is rare and can be confused with \(\varphi(3,0)\) in Veblen's function. It can intuitively by denoted by \(\vartheta(\Omega\uparrow\uparrow\omega)\) which may be useful for beginners, despite \(\Omega\uparrow\uparrow\omega\) being an ill-defined term.

An early version of Bird's array notation was limited by \(\vartheta(\varepsilon_{\Omega+1})\).


  1. Jäger, Gerhard. Die konstruktible Hierarchie als Hilfsmittel zur beweistheoretischen Untersuchung von Teilsystemen der Mengenlehre und Analysis. na, 1979.
  2. Pohlers, Wolfram. Proof theory: The first step into impredicativity. Springer Science & Business Media, 2008.
  3. Michael Rathjen, "Fragments of Kripke–Platek Set Theory with Infinity" (a survey without a proof or a reference to the first source)
  4. J. Ven der Meeren, M. Rathjen, A. Weiermann, An order-theoretic characterization of the Howard-Bachmann-hierarchy (p.1)


  1. Ordinal analyses of set theory was first established by Jäger. Jäger defined proof-theoretic ordinal for set theory, and analysed \(\mathsf{KP}\omega\). This history is stated on the Pohlers' book cited above.

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)