User blog comment:Edwin Shade/Rank-on-Rank Turing Ordinals and Beyond/@comment-26454151-20180120005845/@comment-32876686-20180120021028

You have brought out a good point, likewise, by defining \(\omega[n]=\Sigma(n)\) it appears \(f_{\omega_1^{CK}(n)<^*f_{\omega}\). There is a way I have to counter this however. Only systems of fundamental sequences for an ordinal \(\gamma<\Omega\) such that \(f_{\alpha}(n)>^*f_{\beta}(n)\), where \(\alpha>\beta\) will be allowed, where \(>*\) is the eventual domination symbol.