Small Veblen ordinal

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 \(\varphi(\omega,0)\).I. Lepper, G. Moser, Why ordinals are good for you (p.2). Accessed 2021-06-16.
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.T. Kihara, Lower bounds for tree(4) and tree(5), とりマセ Σ^0_2, 05/2020.
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.Michael Rathjen, "Fragments of Kripke–Platek Set Theory with Infinity" (a survey without a proof or a reference to the first source)
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,Bird, Chris. Beyond Bird’s Nested Arrays II. and {n,n [1 [1 \ 1,2 ~ 2] 2] 2} in the Hierarchial Hyper-Nested Array Notation.Bird, Chris. Beyond Bird’s Nested Arrays IV.
In the slow-growing hierarchy, \(g_{\vartheta(\Omega^{\omega})}(n)\) is approximately {n,n (1) 2} in BEAF.
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== See also ==
ja:小ヴェブレン順序数
zh:小凡勃倫序
Category:Transfinite ordinals