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

I have previously made the comment that functions of the form phi(1,phi(2,0)) or e_{zeta_0} are not well defined. The FS on this blog avoids them and my blog on extended normal form, is my attempt to construct ordinals that do not have this form.

Therefore, the equation in littlepeng's comment

phi(1,phi(2,0)+1)=phi(2,0)^^w=phi(2,1)=phi(1,phi(2,1)) creates its own inconsistenccies.

My FS on this blog simply states phi(2,0)^^w=phi(2,1) and does not allow veblen functions of the form phi(1,phi(2,0)+1), or, phi(1,phi(2,1)).

I would like to make a related argument, and I may need to start a new blog to outline the argument. For the moment, the argument runs along these lines:

a.  it is not possible to have a veblen function of the form phi(a,b) where b = phi(c,d) and c > a and d>0.

b.  constructing an ordinal of that form is not possible

c.  start with phi(a,0) and progressively increment to larger ordinals

d. we reach phi(a,phi(a,0)) then phi^3(a,0_*) and eventually phi^w(a,0_*) = phi(a+1,0)

therefore max(b) = phi^{w-1}(a,0_*) = phi^w(a,0_*) =  phi(a+1,0)

I need to flesh this out further.