User blog comment:Vel!/1+w/@comment-2033667-20141011215630/@comment-5982810-20141011222139

Because in a sense ... they are all ordinals. It's just that the system would be assigning multiple fundamental sequences to the same ordinal. Look every one of these expressions can be proven to match up with a particular ordinal if we assume that

a = a[0] U a[1] U a[2] U ...

Under that consideration 1+w and w are the same "ordinals". But what am I really assigning fundamental sequences to? It must be ''ordinal expressions. ''ie. every so called OLS is a perfectly legimate expression of ordinal arithmetic. But each is treated as a distinct index, instead of relating them back to ordinals. This means I no longer have to define ''canonical forms. ''The downside is, it has to be shown that every OLS is mappable to an ordinal so that the system as a whole is well-ordered. But since every OLS is just a valid expression of ordinal arithmetic, then everyone also maps to an ordinal, and the system must still be well ordered.

To be fair though, I guess I need to clarify this point in the article. To be honest, when I first wrote the article, I did not consider these fine points. It seemed clear to me that 1+w = w and (n)[w] != (n)[1+w]. I didn't see any contradiction there. Now I at least can see that since I say the function is (n)[a] where "n" is a number and "a" is an ordinal, it's problematic, since equalities of ordinal arithmetic should follow. So I do have to change what "arguments" the function is taking. I think the correction interpretation is that it takes a number, and not an "ordinal-like string", but an expression of ordinal arithmetic. Or if that's still too problematic, just admit that they are just indexes within an index set, which take on the forms of ordinals, to take advantage of some established notation and theory. I don't think there is anything disfunctional with that. The only problem with it seems to be extra-mathematical considerations of confusing the googology community.

Thoughts?