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 Weiermann's theta function, it can be expressed as \(\vartheta(\Omega^\omega)\).

Using the Veblen hierarchy as extended to finitely many arguments, it is equal to \(\varphi_{\Omega^\omega}(0)\)

Harvey Friedman's tree(n) function (for unlabeled trees) grows at around the same rate as \(f_{\vartheta(\Omega^\omega)}(n)\) in the fast-growing hierarchy.

It is the growth rate of {X,X,1,2} & n in BEAF, and all higher structures are well-formalised in the sense "if these structures will not rely on lower structures, then they are well-defined".