User blog comment:P進大好きbot/A larger orderinal does not necessarily correspond to a greater function through FGH/@comment-10262436-20190326225307/@comment-35470197-20190326232901

Hi. My counterexample above is \(\omega^{\omega}+1\) equipped with a specific (unusual) system of fundamental sequences. Therefore it occurs even though we consider ordinals below LVO.

Fortunately, many known systems of fubdamental sequences satisfy the desired property, i.e. \(\beta < \gamma\) implies \(f_{\beta} < f_{\gamma}\), and hence we do not have to be careful when we use them. On the other hand, when you define your own system of fundamental sequences, then the problem possibly occurs.

This phenomenon more likely occurs when you use other hierarchy such as SGH or HH. For example, "analysis" of BMS using ordinals seems very likely to include this critical trouble, while few analysts take care (maybe because they are not unaware of it).