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[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 $$\eta_0$$, 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 $$\vartheta(\varepsilon_{\Omega+1})$$, and in Hierarchial Hyper-Nested Array Notation, $$f_{\vartheta(\varepsilon_{\Omega+1})}(n)$$, it is approximately {n,n [1 [1 ¬ 3] 2] 2}.

## Sources

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)
5. Beyond Nested Arrays III (Retrieved 25 April, 2022)
6. Beyond Nested Arrays IV (Retrieved 25 April, 2022)

## Footnotes

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.