User blog:PsiCubed2/Something to Ponder Regarding Sequences and Ordinal Notations

Consider an arbitrary large countable ordinal α.

And suppose that we already have fundamental sequences for every limit ordinal β≤α.

As many of you already know, given this information, we can write any ordinal γ<α less than α as a finite sequence of integers that represent the FS's path from α to γ.

For example, let's say that we want an ε₀-level notation. We'll set α to ε₀ and use the Wainer Hierarchy of fundamental sequences. Then we'll have:

(1,1) = (1) = ε₀[1] = 1

(1,2) = ε₀[2][2] = ω[2] = 2

(1,3) = ε₀[2][3] = ω[3] = 3

(1,4) = ε₀[2][4] = ω[4] = 4

(2,1) = (2) = ε₀[2] = ω

(2,1,1,1) =  ε₀[3][2][2][1] = [ωω][2][2][1] = [ω2][2][1] = [ω×2]1 = ω+1

(2,1,1,2) =  ε₀[3][2][2][2] = [ωω][2][2][2] = [ω2][2][2] = [ω×2]2 = ω+2

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(2,1,2) =  ε₀[3][2][2] = [ωω][2][2] = [ω2][2] = ω×2

(2,2) = ε₀[3][2] = [ωω][2] = ω2

(2,3) = ε₀[3][3] = [ωω][3] = ω3

(3) = ε₀[3] = ωω

and so on.

Note that a similar system exists for any countable ordinal. This is because the fundamental sequences are always guaranteed to exist, and the process (for any given countable ordinal) is guaranteed to terminate.

Now, so far, I haven't really done anything new. The point I want to make in this post is this: If we have an ordinal that we only have partial information on, we can construct part of the system that corresponds to that ordinal.

Say we set α to PTO(Z2). While we don't know how to "count" up to PTO(Z2), we do know of a fundamental sequence of PTO(Z2) itself:

PTO(Z2) = limit (1, PTO( Π11−CA0), PTO( Π12−CA0),  PTO( Π13−CA0) ... )

So we can set:

(1) = 1

(2) = PTO( Π11−CA0)

(3) = PTO( Π12−CA0)

(4) = PTO( Π13−CA t0)

Now, we also know the PTO( Π11−CA0) is ψ(Ωω). So all the arrays that start with (1) should correspond to ordinals smaller than this. Say:

(1,1) = BHO

(1,2) = ψ(ψ2(0))

(1,3) = ψ(ψ3(0))

and so on.

We can also easily fill in the blanks between (2) and (3) even if we don't know any specific fundmanetal sequence of PTO( Π12−CA0). Since a fundamental sequence is infinite, we can add any (finite) number of arbitrary terms at the start and it won't change the limit. So let's just right down a few large ordinals that would make a nice progression:

(2) = PTO( Π11−CA0) = ψ(Ωω)

(2,1) = ψ(ΩΩ)

(2,2) = ψ(I)

(2,3) = ψ(M)

(2,4) = ψ(K)

(2,5) = ψ(T) (or whatever)

Of-course we'll need to use an OCF that is actually well-defined for this to work as an actual notation, but even that isn't necessary if all we won't is to illustrate how it might be done.

Note that we are free to add as much detail as we want, and we'll never "ruidn out" of space. Note also, that the partial notation we have at any given time makes complete sense. In fact, a person with a full knowledge of how PTO(Z2) works, could turn our outline into a fully-fledged Z2-level ordinal notation with no gaps(*).

The interesting thing here is, that he would be able to do that regardless of our choices. Even if we actively tried to botch the notation, there would always remain an infinite amount of space to correct these errors.

Which leads us to this amazing conclusion: As long as our work is limited to a finite list of examples, we haven't progressed a single inch toward creating a full-fledged ordinal notation.

Now, does all the above remind you of anything you've seen here recently?

(*) It is also equally easy to extend such a notation to a larger ordinal. We simply insert PTO(Z2) at the beginning of the fundamental sequence of that ordinal. Now the ordinal that was represented as (say) (4,6,2), will now be represented as (1,4,6,2). PTO(Z2) itself, of-course, would be (2) in our new system, and ordinals that start with (2) will now be bigger than PTO(Z2).