User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-35392788-20191031140942

Would it work to require \(\beta\) to be fixed for all \(n\) ? It definitely works in your counterexample, since \(\omega^n+\omega2\) is obviously fixed. If so, then the change might be this :

\(\forall \beta\exists n\forall\alpha>\beta\forall m\geq n\exists k\beta\) such that at least n elements of the fundamental sequence of \(\alpha\) belong to \(\beta\).

It seems a bit easy, but it looks like it works for your counterexample. Note that the system presented in your blog post still does not respect the modified condition, because \(\beta=\omega\) is indeed fixed.