User blog comment:P進大好きbot/Proposal to Choose an Official Standard OCF in This Wiki/@comment-11227630-20191219030804/@comment-35470197-20191219091845

The base function Ω is not necessary if I correctly understand it. The base function φ is often used in order to strengthen the limit. For example, Rathjen's OCF based on M uses it in order to make ε_{M+1} a portion of the system, and Rathjen's OCF based on K uses it in order to make ε_{K+1} a portion of the system.

I note that what I refered to is not mainly base function, but conditions on the constructibilities and other subtools such as α^* and κ^- in Rathjen's OCF based on M. They are frequently deleted in "simplications" by wiki users, while they play importatnt roles in order to construct ordinal notations.