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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, $$g_{\vartheta(\Omega^{\omega})}(90)$$ is approximately Goobolspeck.

## Sources

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.