User blog comment:Testitemqlstudop/FPT 2/@comment-34193315-20191123002346

It's really hard to understand what exactly you're defining, so please correct me if I'm wrong.

If I understand everything:

\(f_0(0)  \) is a function, \( f_0(0)(\alpha)=\varepsilon_\alpha \)

\(f_0(1) \) is the nth fixed point of \(f_0(0) \), so \(f_0(1)=\zeta_\alpha \)

\(f_0(n)(\alpha) = \phi(1+n,\alpha) \)

\(f_0(\omega) \) is now undefined, because you didn't define the function for limit ordinals. (Try fixing this first)

After you fix that, maybe what you wanted was \( f_0(\omega)(\alpha) = \phi(\omega,\alpha) \).

Eventually you would have \(f_0(\varepsilon_0) is the function \alpha \mapsto \phi(\varepsilon_0,\alpha) \)

I know you want to nest the first argument of the veblen function, so let's just assume that you done that already.

\(f_1(0)(\alpha)=\Gamma_0[\alpha] \) and I think you intended this. Even though you will run into problems for limit ordinals, but let's assume you fixed this as well.

Then you have something like \(f_n(\beta)(\alpha)=\phi(n,\beta,\alpha) \). Again you will have some problems for limit ordinal n.

So T should be the Ackermann ordinal, \( \psi(\Omega^{\Omega^2}) \).