User blog comment:B1mb0w/Fundamental Sequences/@comment-1605058-20151104112933/@comment-10262436-20151104125521

but we don't want different values for the fundamental sequence of an ordinal... we need to define away any ambiguity. In the case g = w^2+w, we need to assert the fundamental sequence for g to be precisely w^2+w and never (w+1).w The definition will be something along these lines.... 1. express g in terms of limit ordinals. 2. group all powers of limit ordinals together in the form beta^p.m, e.g. w^p.m 3.  order all powers of limit ordinals in descending order 4. recursively apply these rules to p and m as appropriate 4. the smallest power of the smallest limit ordinal (i.e. the last term) is then acted on to determine the next element in the fundamental sequence