User blog comment:Eners49/A whole new superclass of infinities?/@comment-3427444-20180722052905/@comment-30754445-20180727090200

It wasn't Syst3m who proposed this, but me.

And yes, you got the gist of the idea.

In theory, it is possible to define this in such a way that fω(ω) = φ(ω,0). It's not guaranteed, though. Indeed, if you try to extend hyper-operators to ordinals in the most straightforward way, your system will get stuck at ε₀ (which, as you may recall, is equal to φ(1,0)).