 Though the “great Veblen number” (as \(E(0)\) is sometimes called) is quite an impressive ordinal it does not furnish an ordinal representation sufficient for the task of analyzing a theory as strong as \(\Pi^1_1\)-comprehension. Of course, it is possible to go beyond \(E(0)\) and initiate a new hierarchy based on the function \(\xi\mapsto E(\xi)\) or even consider hierarchies utilizing finite type functionals over the ordinals. Still all these further steps amount to rather modest progress over Veblen’s methods. In 1950 Bachmann presented a new kind of operation on ordinals which dwarfs all hierarchies obtained by iterating Veblen’s methods. Bachmann builds on Veblen’s work but his novel idea was the systematic use of uncountable ordinals to keep track of the functions defined by diagonalization. Let \({\Omega}\) be the first uncountable ordinal. Bachmann defines a set of ordinals \(\fB\) closed under successor such that with each limit \(\lambda\in \fB\) is associated an increasing sequence \(\langle \lambda[\xi]:\,\xi&lt;\tau_{\lambda}\rangle\) of ordinals \( \lambda[\xi] \in \fB\) of length \(\tau_{\lambda}\leq\Omega\) and \(\lim_{\xi&lt;\tau_{\lambda}}\lambda[\xi]=\lambda\). A hierarchy of functions \((\varphi^{\ssfB}_{\alpha})_{\alpha \in \fB}\) is then obtained as follows: