Bachmann-Howard ordinal

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 infinityJäger, Gerhard. Die konstruktible Hierarchie als Hilfsmittel zur beweistheoretischen Untersuchung von Teilsystemen der Mengenlehre und Analysis. na, 1979.Pohlers, Wolfram. Proof theory: The first step into impredicativity. Springer Science & Business Media, 2008.Michael Rathjen, "Fragments of Kripke–Platek Set Theory with Infinity" (a survey without a proof or a reference to the first source)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.. 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 \(\eta_0\)J. Ven der Meeren, M. Rathjen, A. Weiermann, An order-theoretic characterization of the Howard-Bachmann-hierarchy (p.1)
, 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 \(\vartheta(\varepsilon_{\Omega+1})\),Beyond Nested Arrays III (Retrieved 25 April, 2022) and in Hierarchial Hyper-Nested Array Notation, \(f_{\vartheta(\varepsilon_{\Omega+1})}(n)\) is approximately {n,n [1 [1 ~ 3] 2] 2}.Beyond Nested Arrays IV (Retrieved 25 April, 2022)
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zh:巴赫曼-霍華德序
ja:バッハマン・ハワード順序数
Category:Transfinite ordinals
Category:Buchholz's psi-functions