User blog comment:B1mb0w/Fundamental Sequences/@comment-10262436-20160206105120/@comment-5529393-20160206113328

Yes, you have made comments like that and they don't make any sense. e_{zeta_0} is perfectly well-defined, it's zeta_0. Is your problem that we have a function with a fixed point? There are lots of functions with fixed points, for example if f(x) = x^2 then f(1) = 1. Nothing wrong with that.

phi(2,)^^w = phi(2,1) is not true; if you wish to define your own function, that's perfectly fine, but please do not call them Veblen functions and say that there is something wrong with the original version, unless you actually find a problem with the original version.

You points a. and b. are not literally true, it is indeed possible to have a phi(a,phi(c,d)) with c > a and d > 0. Why would it not be? In fact if I have some function f from a set X to itself, I can always take f(a, f(c,d)) for any a,c,d in X. How could it be otherwise? You need to provide some reasoning for your assertions.

Anyways, you can define your own function however you want, but the way you are doing it it's going to be much weaker than the standard Veblen function.