User blog comment:Moooosey/Probably Unnotable Stuff by an Amateur Googologist/@comment-35470197-20190927224722

I note that FGH along ordinals and fixed fundamental sequences is uncomputable in the sense that it lack an effective algorithm to compute the values. (It can be computable in the sense that there can be a Turing machine immitating the function.) Uncomputable functions are allowed in googology, but many of uncomputable large functions are much bigger than functions given by FGH along ordinals and fixed fundamental sequences. Therefore googologists tend to choose either to make functions computable or to create uncomputable functions much stronger than FGH.

In order to avoid the uncomputability, googologists use notations equipped with fundamental sequences.