User blog comment:Vel!/1+w/@comment-5982810-20141011211100

" I apologize for the hostile and abrasive tone of this post, but I'm baffled at (and fed up with) the amount of misconceptions around these concepts." -- Vel

I realize that my arguing here can lead many less informed individuals to the wrong conclusions. As I already noted in my articles on FGH, 1+w and w can be defined as the same set since I define the "set" of any index as the union of all the indexes of it's fundamental sequences (although this is entirely unneccessary and has nothing to do with evaluating the function). This shows that 1+w and w are representing the same "ordinal". The main significance of this is that if we have indexes i and j, and their fundamental sequences have the same supremum (ie. they are the same ordinal), then (n)[ i ] will grow at the same rate as (n)[ j ], but they will not be the same function. They have distinct fundamental sequences. How do you know which one to chose if i = j ? Based on the INDEX in the function, not the ordinal it represents. ie. based on the form not it's content. I never claimed 1+w != w. I only claim that (n)[1+w] != (n)[w] in my variant.