User blog comment:Pellucidar12/Attempt at a FGH related notation/@comment-29915175-20170416151157/@comment-1605058-20170418193600

I don't quite see what your point is. What exactly do you consider to be a "drawback"? We can't prove anything of that sort if we don't know what we are talking about.

I, personally, don't see any problem with \(\omega\) having this fundamental sequence and not a different one; in a way all fundamental sequences are just as good or bad. If you do want some ordinals to have the FSes you want, then feel free to define FSes up to some point (like \(\varepsilon_0\)) one way and above it another way.

And when it comes to intuition - not sure about you, but for me it does make intuitive sense that any increasing sequence of natural numbers will be fine for an FS of \(\omega\).