User blog comment:Deedlit11/My humble extension of FOOT/@comment-5529393-20170117081155/@comment-27513631-20170122102658

I think blog posts tend to get buried after a week. At least, I no longer look for replies. :P

Anyway, I don't entirely agree with what you've said. It seems to me that \(Func^1_\alpha(\gamma)\) is the \(\gamma\)th correct cardinal in the language \(\{\in,(Func^1_\delta)_{\delta<\alpha}\), and hence corresponds to \(T(S,\alpha,\gamma)\) for some \(S\) defining the corresponding correctness. I do still, however, agree that \(FOFT^2_0\) is enough to beat Lil'Bigeddon.

Also, you've got the \(f_\gamma\) as separate parameters, so only finitely many of them can be used in a given statement. I think that correcting that would make \(FOFT^\omega_0\), \(FOFT^{\omega_1}_0\) and similar considerably stronger.