11,582
pages

The large Veblen ordinal is a large countable ordinal. In the Veblen hierarchy as extended to transfinitely many arguments (also called Schutte's Klammersymbolen), it is the least fixed point of the ordinal function $$\alpha \mapsto \varphi \left( \begin{array}{c} 1 \\ \alpha \end{array} \right)$$. Using Weiermann's $$\vartheta$$ function, it can also be denoted $$\vartheta(\Omega^\Omega)$$ and is the first fixed point of $$\alpha \mapsto \vartheta(\Omega^\alpha)$$. Additionally, using Madore's $$\psi$$ function or Buchholz's $$\psi$$ function, it is equal to $$\psi_0(\Omega^{\Omega^\Omega})$$. This ordinal has also been called the "Great Veblen number" and denoted as $$E(0)$$[1].

Jonathan Bowers has mentioned "LVO-order set theory" while discussing hypothetical ways to beat Rayo's number without defining it.[2]

In the fast-growing hierarchy, 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, and {n,n [1 [1 \ 1 \ 2 ¬ 2] 2] 2} in the Hierarchial Hyper-Nested Array Notation.[citation needed]

## Sources

1. M. Rathjen, The art of ordinals analysis (p.55, 2006, accessed 2020-11-11)
2. Bowers, Jonathan. Going to Oblivion. Retrieved 2016-12-14.