User blog comment:B1mb0w/Fundamental Sequences/@comment-5529393-20160124015439/@comment-10262436-20160206081857

Lets discuss phi(2,1) as the relevant example for these comments. I claim:

phi(2,1) = phi(2,0)^^w (as per rule 2B)

You have stated phi(2,0)^^w =phi(1,phi(2,0)+1) in your comments. But as I pointed out in my previous comment, this equivalence seems to add no explanatory power because it seems to imply phi(2,0) = phi(1,phi(2,0)) and gets us no closer to understanding phi(2,1). In other words the equation you have stated is self-evident. It does not prove or disprove my claim.

I am being careful to clearly define the fundamental sequences I am using and I am not claiming my FS is anything more than one of many FS that exist. However, it seems clear to me that the choice of FS will either change the value of the veblen function, or, change the value of a diagonalised veblen function for [n] where n is finite. This has come up many times in comments where zeta_0[2] could equal e_{e_0} or e_{e_1} or other values based on the cchoice of FS.

I hope this is making my position clearer.