User blog:ArtismScrub/Hierarchical Cardinal Notation (HCN)

So, I'm sure people have had this idea before.

Take the Veblen hierarchy and replace the base rule φ(0,α) = ωα with Φ(0,α) = Ωα.

(I use capital phi analogous to ω vs. Ω.)

In fact, this can continue further than Φ(Ω,0) WITHOUT having to collapse the uncountable ordinals, since it'll be forming uncountable ordinals to begin with.

However, there will still exist a fixed point of α ↦ Φ(α,0), most certainly. So, obviously, the next step would be to use Φ(1,0,0)--since Ω is valid in its literal form, there is nothing to use as a collapsing ordinal...

or is there?

Enter the inaccessible cardinal, 𝐈. Φ(𝐈,0) is the first fixed point of α ↦ Φ(α,0). Similarly to any OCF, this can continue with 𝐈+1, 𝐈×2, 𝐈2, 𝐈𝐈, ε𝐈+1... 𝐈2, 𝐈ω, 𝐈Ω, 𝐈𝐈, 𝐈𝐈 𝐈 𝐈 ... and so on.

Now we need a notation to describe the inaccessible cardinals.

Φ2(0,α) = 𝐈α, all other rules remain normal.

Yes, a second order function. We can say that Φ0(α,β) would just be the normal two-argument Veblen OCF, Φ1(α,β) will be the Ω-iterator mentioned earlier, and so Φ2(α,β) will be one to iterate inaccessible cardinals.

So what cardinal should be used to collapse on THIS function? Well, I'm no expert on cardinals, but I think the next step up is 𝐌, the Mahlo cardinal. These can then be iterated using:

Φ3(0,α) = 𝐌α, all other rules remain normal.

I'm not sure which cardinal comes next, but it's easy to define a sequence of cardinals designed to collapse on and ennumerate the fixed points of the previous. So, this can continue with Φ4(α,β), Φω(α,β), ΦΩ(α,β), etc. In fact, even if the cardinals I've described are ill-defined, they can still work merely as symbols with which to iterate each function like an OCF.

The limit of all this is the fixed point of α ↦ Φα(0,1), an EXTREMELY large cardinal.

Of course, to prevent having to write out transfinitely many symbols just for "Φ(Φ2(Φ3(Φ4(...,0),0),0),0)", we can declare that Φ(Φα(β,γ),δ) = Φ(Φ2(Φ3(Φ4(...Φα(β,γ)...,0),0),0),δ).

So tell me: Has this been done before? Is this properly defined? How would this be defined using practical ordinal language? And, most importantly, how strong would this notation at its limits be if plugged into an OCF?