I've read and known what the FGH is all the way up to a>Bachmann Howard ordinal which is ψ(Ω_2), the Googology Wiki, Wikipedia, Chris Bird articles, and YouTube don't show how the Fast Growing Hierarchy works beyond this. Can someone please tell me how say a subcubic number (which I define as ψ(Ω_ω)).
The Googology Wiki says that still this is not an ill defined OCF, can someone please explain how ordinals beyond BHO work in-depthly?
The problem is that the fast-growing hierarchy isn't defined for ordinals but rather ordinal notations: if you want to continue it up to some ordinal $\alpha$ you need to fix a notation system for all ordinals $<\alpha$, and different notation systems below the same ordinal can yield very different fast-growing hierarchies along that ordinal so this really isn't a step we can skip past. That said, there's nothing mysterious about Bachmann-Howard - if you give me a notation system extending beyond it, I'll give you a fast-growing hierarchy going past it to match.
Now you might reasonably ask why we don't just fix a notation system for all countable ordinals at once - or at least, for all computable ordinals. The answer is that there is simply no "concrete" way to do so. For example, it's consistent with ZF (= set theory without the axiom of choice) that there is no function sending each countable ordinal $\alpha$ to a well-ordering of a set of natural numbers with ordertype $\alpha$. Similarly, there's no partial computable function $f$ such that $(i)$ $dom(f)\supseteq\mathcal{O}$ and $(ii)$ for $m,n\in\mathcal{O}$ we have $f(m)=f(n)$ iff $\vert m\vert_\mathcal{O}=\vert n\vert_\mathcal{O}$. (The latter point is easier to prove but harder to understand in my opinion.)
