User blog comment:Alejandro Magno/SPECIAL! FGH of Hyperion Notation/@comment-5982810-20140926214442

First off, Alejandro is only using 1+w in his analysis because I did first. It is not correct to say (0)#n = f_w(n). Rather, we find that (0)#n = n#n = f_1+n(n). Using the following fundamental sequence for 1+w = {1,2,3,...} we can define this as f_1+w(n).

Second, I don't understand what the big problem is here. Saying f_1+w(n) != f_w(n) has nothing to do with violating the axiom of extensionality. I'm not saying that 1+w is a different ordinal/set than w. An ordinal can be defined as the union of it's fundamental sequence. The union of {1,2,3,...} is the same as the union of {0,1,2,...}, thus 1+w and w are the same set. None the less the difference shows up if we chose different members from the same set. In this way ... we may think of 1+w not so much as an ordinal but as a label for a ''fundamental sequence. 1+w and w are not distinct ordinals, but they are distinct fundamental sequences for the ordinal w.  Key thing being  there is not one unique fundamental sequence for any given ordinal, therefore there is no necessity to have one unique form for an ordinal, it's purely a convention.''

There is no need to replace rogue-types with canonical forms. They have well defined fundamental sequences that are distinct from the fundamental sequences of canonical forms.

Bottom line: We define notation systems, NOT ordinal systems. They may have close relations to ordinals, but this doesn't mean every property of ordinals and ordinal arithmetic carries over. There is nothing ill-defined in this approach. Each member of the set will have a fundamental sequence that can be selected from. The only thing you could reasonably argue is that there may exist non-terminating sequences for some rogue-types. At very least this is not the case for 1+w, because 1+w immediately decays to a canonical form. Furthermore 1+ a canonical form must decay to a canonical form because ordinal addition is worked out from right to left, and we only begin on the next term in the sum once the last term has been reduced to 0. The 1+ can only be dealt with once it's second part is NOT a limit ordinal, ie. {0,1,2,3,...}, a member of w. In all other respects the sequence for 1+a will look identical to "a" except with a 1+ in the beginning.