User blog:ArtismScrub/Hyper-Alpha Notation, Part IV Redone

(WORK IN PROGRESS)

See:

Part I

Part II: Arrows to Alphas

Part III: Full Circle

Original Part IV, a stupid attempt at an extension that will be ignored for this purpose

Let's get down to business. In the original Part IV, I defined <A,A> as an operator and allowed for operations on <A,A>. That sucked. I have a better idea.

A₁ = A

<a,b>★A_n = <a,a>★<A_n-1,A_n-1>^b (exponent denotes concatenation, NOT an operator)

A_n will still evaluate to <A_n-1,A_n-1>^b if placed after any number of up-arrows.

Definition may change.

Recursive progression

<a,b>A₂ ≈ ψ(Ω_ω)

<a,b>A₂A ≈ ψ(Ω_ω)+1

<a,b>A₂A₂ ≈ ψ(Ω_ω)2

<a,b>A₂^A ≈ ψ(Ω_ω)×ω

<a,b>A₂^A↑↑A ≈ ψ(Ω_ω)×ε₀

<a,b>A₂^A₂ ≈ ψ(Ω_ω)²

<a,b>A₂^A₂A ≈ ψ(Ω_ω)^ω

<a,b>A₂^A₂A↑↑A ≈ ψ(Ω_ω)^ε₀

<a,b>A₂↑↑3 ≈ ψ(Ω_ω)↑↑2

<a,b>A₂↑↑n ≈ ψ(Ω_ω)↑↑(n-1)

<a,b>A₂↑↑A ≈ ψ(Ω_ω+Ω)

<a,b>A₂↑↑A↑↑A ≈ ψ(Ω_ω+Ω2)

<a,b>A₂↑↑A↑↑↑A ≈ ψ(Ω_ω+Ω²)

<a,b>A₂↑↑A↑↑↑↑A ≈ ψ(Ω_ω+Ω³)

<a,b>A₂↑↑A ↑^A A ≈ ψ(Ω_ω+Ω^ω)

<a,b>A₂↑↑<A,A>A^AA ≈ ψ(Ω_ω+Ω^Ω)

<a,b>A₂↑↑<A,A>A↑↑A ≈ ψ(Ω_ω+Ω₂)

<a,b>A₂↑↑<A,A><A,A>A^AA ≈ ψ(Ω_ω+Ω₃)

<a,b>A₂↑↑A₂ ≈ ψ(Ω_ω×2)

<a,b>(A₂↑↑A₂)↑↑A ≈ ψ(Ω_ω×2+1)

<a,b>(A₂↑↑A₂)↑↑A↑↑A ≈ ψ(Ω_ω×2+Ω)

<a,b>(A₂↑↑A₂)↑↑A₂ = A₂↑↑A₂A ≈ ψ(Ω_ω×3)

<a,b>A₂↑↑A₂A^A ≈ ψ(Ω_ω×ω)

<a,b>A₂↑↑A₂(A↑↑A) ≈ ψ(Ω_ω×Ω)

<a,b>A₂↑↑A₂A₂ ≈ ψ(Ω_ω²)

<a,b>A₂↑↑A₂^A ≈ ψ(Ω_ω^ω)

<a,b>A₂↑↑A₂^A↑↑A ≈ ψ(Ω_ω^Ω)

<a,b>A₂↑↑A₂^A₂ ≈ ψ(Ω_ω^Ω_ω)

<a,b>A₂↑↑A₂↑↑A ≈ ψ(Ω_ω+1)

<a,b>A₂↑↑A₂↑↑A^A ≈ ψ(Ω_ω2)

<a,b>A₂↑↑A₂↑↑A↑↑A ≈ ψ(Ω_Ω)

<a,b>A₂↑↑A₂↑↑A↑↑↑A ≈ ψ(Ω_Ω²)

<a,b>A₂↑↑A₂↑↑<A,A>A^AA ≈ ψ(Ω_Ω2)

<a,b>A₂↑↑A₂↑↑A₂ ≈ ψ(Ω_Ωω)

<a,b>A₂↑↑A₂↑↑A₂↑↑A ≈ ψ(Ω_Ωω+1)

<a,b>A₂↑↑A₂↑↑A₂↑↑A₂ ≈ ψ(Ω_ΩΩω)

<a,b>A₂↑↑↑A ≈ ψ(OFP)

I don't know how to express ordinals beyond ψ(OFP) using an efficient notation. Until then I will use capital epsilon (E) to denote the αth omega fixed point as E_α, and continue using the Veblen hierarchy but with capital Greek letters. This is most likely ill-defined and should be left up to interpretation.

<a,b>A₂↑↑↑A ≈ ψ(E₀)

<a,b>(A₂↑↑↑A)↑↑A₂ ≈ ψ(Ω_E0)

<a,b>A₂↑↑↑AA ≈ ψ(E₁)

<a,b>A₂↑↑↑A^A ≈ ψ(ψ_I(E_ω)

<a,b>A₂↑↑↑A↑↑A ≈ ψ(E_ε0)

<a,b>A₂↑↑↑A↑↑↑A ≈ ψ(E_ζ0)

<a,b>A₂↑↑↑A ↑^A A ≈ ψ(E_φ(ω,0))

<a,b>A₂↑↑↑<A,A>A^AA ≈ ψ(E_ψ(Ω2))

<a,b>A₂↑↑↑A₂ ≈ ψ(E_ψ(Ωω))

<a,b>A₂↑↑↑A₂A ≈ ψ(E_ψ(Ωω)+1)

<a,b>A₂↑↑↑A₂↑↑↑A ≈ ψ(E_ψ(E0))

<a,b>A₂↑↑↑↑A ≈ ψ(E_Ω), the first fixed point of α ↦ ψ(E_α)

<a,b>(A₂↑↑↑↑A)↑↑↑A₂ ≈ ψ(E_Ω+1)

<a,b>A₂↑↑↑↑AA ≈ ψ(E_Ω+ω)

<a,b>A₂↑↑↑↑A↑↑A ≈ ψ(E_Ω+ε0)

<a,b>A₂↑↑↑↑A₂ ≈ ψ(E_Ω+ψ(E0))

<a,b>A₂↑↑↑↑A₂A^A ≈ ψ(E_Ω2)

<a,b>A₂↑↑↑↑A₂A₂ ≈ ψ(E_Ω×ψ(E0))

<a,b>A₂↑↑↑↑A₂^A ≈ ψ(E_Ω²)

<a,b>A₂↑↑↑↑A₂↑↑↑A ≈ ψ(E_E0)

<a,b>A₂↑↑↑↑A₂↑↑↑↑A ≈ ψ(E_EΩ)

<a,b>A₂↑↑↑↑↑A ≈ ψ(Z₀)

<a,b>A₂↑↑↑↑↑AA ≈ ψ(Z₁)

<a,b>A₂↑↑↑↑↑A₂ ≈ ψ(Z_ψ(Ωω))

<a,b>A₂↑↑↑↑↑A₂A^A ≈ ψ(Z_Ω)

<a,b>A₂↑↑↑↑↑A₂↑↑↑A ≈ ψ(Z_E0)

<a,b>A₂↑↑↑↑↑A₂↑↑↑↑↑A ≈ ψ(Z_Z0)

<a,b>A₂↑↑↑↑↑↑A ≈ ψ(H₀)

<a,b>A₂↑↑↑↑↑↑A₂ ≈ ψ(H_ψ(Ωω))

<a,b>A₂↑↑↑↑↑↑↑A ≈ ψ(Φ(4,0))

<a,b>A₂ ↑^A A ≈ ψ(Φ(ω,0))

<a,b>A₂ ↑^{A ↑A A} A ≈ ψ(Φ(φ(ω,0),0))

<a,b>A₂ ↑^A₂ A ≈ ψ(Φ(ψ(Ω_ω),0))

<a,b>A₂ ↑^{A₂ ↑A A} A ≈ ψ(Φ(ψ(Φ(ω,0)),0))

As you can see, A₂ is a very powerful extension by itself. But A₃ diagonalizes over it just the same.

<a,b>A₃ ≈ ψ(Φ(Ω,0)), the first fixed point of α ↦ ψ(Φ(α,0)).

WIP article