User blog comment:ArtismScrub/Hierarchical Cardinal Notation (HCN)/@comment-1605058-20180226215934

A problem with the definition as stated is that the Veblen hierarchy rules, once you redefine \(\Phi(0,\alpha)=\Omega_\alpha\), already defines \(\Phi(\alpha,\beta)\) for all ordinal \(\alpha,\beta\). Veblen hierarchy doesn't behave like collapsing functions, and if you want it to behave differently, you have to redefine it from the beginning.