User blog comment:P進大好きbot/Relation between an OCF and an Ordinal Notation/@comment-30754445-20180810102658

Personally, I don't see the problem with the statement "an OCF is an ordinal notation".

Sure, it isn't precise, but the intention clear. What people actually mean by it is "an OCF-based ordinal notation".

And you're right, it isn't easy at all to create a strong OCF-based notation. Making sure that (1) there are no gaps in the countable range, and (2) that the notation is as strong as we want it to be is devilishly difficult.

But it's the most efficient way we currently know to construct large ordinals in a way that allows us to follow the process explicitly. This is required, for example, if we want to create a googological array notation, because we need to know the "expansion rules" at every step of the way.

And this cannot be done with any current notation that reaches PTO(ZFC) or even PTO(Z2). Such a notation is a akin to teleporting yourself to Mt. Everest, instead of actually climbing the mountain. I suppose it still counts for breaking records, but it pretty much sucks the fun out of the game itself.