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 \(\varphi(\omega,0)\).

Harvey Friedman's tree(n) function (for unlabeled trees) is believed to grow at around the same rate as \(f_{\vartheta(\Omega^\omega)}(n)\) in the fast-growing hierarchy with respect to an unspecified system of fundamental sequences.