User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-35470197-20191031095138

> But then I wonder, with respect to the 4 conditions, what would it even mean for the FGH to break ?

For example, I would like to say that FGH associated to a system of fundamental sequences is breaking if there are ordinals/terms \(\alpha < \beta\) such that \(f_{\alpha}\) eventually dominates \(f_{\beta}\). There are several examples of such systems of fundamental sequences satisfying the revised four conditions. For example, I created a system of fundamental sequences up to \(\omega^{\omega}+1\) such that \(f_{\omega^{\omega}\) is weaker than \(f_{\omega+1}\), and professor Kihara created an example of a (non-recursive) system of fundamental sequences up to \(\omega_1^{\textrm{CK}}+1\) such that \(f_{\omega_1^{\textrm{CK}}\) is weaker than \(f_{\omega+3}\).