User blog:ArtismScrub/Hyper-Alpha Notation Part III: Full Circle

See parts 1 and 2.

This is it, the full generalization. I thought you guys said making a functional SGH-FGH map was hard!

Now, as you probably know, I generally do much better explaining notations through progression than through compact recursive rulesets. I have no idea how to put this one entirely into mathematical language. If this notation chokes at any point or the ordinals given are wrong, PLEASE tell me!

Anyway, this of course starts off with:

A ↑A A = A ↑b A

Now, my first idea was to go with ↑A+1 = ↑AA, but shortly afterwards I came up with a much better idea.

A ↑A↑ A.

Bam, A+1-level arrow.

This can continue with:

↑A↑↑ = A+2-level arrow

↑A↑A = A2-level arrow

↑A↑A↑A = A3-level arrow

And, of course, ↑AA = AA-level arrow. Easy!

This essentially means:

↑★A = (↑★)A

Analogous to how:

A★A = (A★)A

Piece of cake.

Of course, even if I were to go with my initial idea, the two would catch up at ↑A↑↑A anyway.

Now, with regards to nested ★: this should seem rather straightforward, with one small issue:

★A = ★b

If b is already A, this turns into a feedback loop.

No problem--just turn (★) into (★) whenever necessary.

Now for some recursive progression. Cue Initial D music, because this one is gonna go by fast:

A ↑A A ≈ φ(ω,0) (this is where we left off last time, also equal to AA)

A ↑A AA ≈ φ(ω,1)

A ↑A AA ≈ φ(ω,ω)

A ↑A (A ↑n A) ≈ φ(ω,φ(n-1,0))

A ↑A (A ↑A A) ≈ φ(ω,φ(ω,0))

A ↑A↑ A ≈ φ(ω+1,0)

A ↑A↑n A ≈ φ(ω+n,0)

A ↑A↑A A ≈ φ(ω2,0)

A (↑A)n A ≈ φ(ωn,0)

A ↑AA A ≈ φ(ω2,0)

A ↑A n  A ≈ φ(ωn,0)

A ↑A A  A ≈ φ(ωω,0)

A ↑A↑↑A A ≈ φ(ε0,0)

A ↑A↑↑↑A A ≈ φ(ζ0,0)

A ↑A ↑ n A  A ≈ φ(φ(n-1,0),0)

A ↑A ↑ A A  A ≈ φ(φ(ω,0),0)

Notice the relationship here: A★ ≈ α, A ↑★ A ≈ φ(α,0). This relationship appears to hold for any ★ ≥ A.

Anyway, continuing:

AAA ≈ Γ0

AAAA ≈ φ(1,1,0)

AAAAA ≈ φ(1,2,0)

AAAn ≈ φ(1,n-1,0)

AAAA ≈ φ(1,ω,0)

AAAAA ≈ φ(2,0,0)

<A,A>(AA)n(Am) ≈ φ(n,m-1,0)

<A,A>AAA ≈ φ(ω,0,0)

<A,A>AAAA ≈ φ(1,0,0,0)

<A,A>AAAAA ≈ φ(1,0,1,0)

<A,A>AAAAA ≈ φ(1,0,ω,0)

<A,A>AAAAAA ≈ φ(1,1,0,0)

<A,A>AAAAAA ≈ φ(1,ω,0,0)

<A,A>AAAAAAA ≈ φ(2,0,0,0)

<A,A>(AAA)nA ≈ φ(n,0,0,0)

<A,A>AAAA ≈ φ(ω,0,0,0)

<A,A>AAAAA ≈ φ(ω,0,0,0,0)

<A,A>AAn ≈ ϑ(Ωnω) or φ(ω,0,0,0,0,...,0,0,0,0) (n "0"s)

<A,A>AAA ≈ ϑ(Ωω)

At this point I'm not even certain on how to work with the ordinal notations in question, so bear with me. I'm judging based off of how <a,b>★ compares to BEAF and the ordinals listed here.

<A,A>AAA A ≈ ϑ(ΩΩ)

<A,A>AAA AA ≈ ϑ(ΩΩ+1)

<A,A>AAA AA ≈ ϑ(ΩΩ+ω)

<A,A>AAA AAA ≈ ϑ(ΩΩ+Ω)

<A,A>AAA AAA ≈ ϑ(ΩΩ+Ωω)

<A,A>AAA AAAA ≈ ϑ(ΩΩ+Ω2ω)

<A,A>AAA AAn ≈ ϑ(ΩΩ+Ωn-1ω)

<A,A>AAA AAA ≈ ϑ(ΩΩ+Ωω)

<A,A>AAA AAA A ≈ ϑ(ΩΩ2)

<A,A>AAA AAA AAA ≈ ϑ(ΩΩ2+Ωω)

<A,A>(AAA )n ≈ ϑ(ΩΩn-1+Ωω)

<A,A>AAA A ≈ ϑ(ΩΩω)

<A,A>AAAA A ≈ ϑ(ΩΩ+1)

<A,A>AAAAA  ≈ ϑ(ΩΩ+1ω)

<A,A>AA AAA A ≈ ϑ(ΩΩ+2)

<A,A>AA AAn  ≈ ϑ(ΩΩ+n-1ω)

<A,A>AA AAn A ≈ ϑ(ΩΩ+n)

<A,A>AA AAA  ≈ ϑ(ΩΩ2)

<A,A>AA AAAAA  ≈ ϑ(ΩΩ3)

<A,A>AA AA  ≈ ϑ(ΩΩω)

<A,A>AA AA A ≈ ϑ(ΩΩ 2 )

<A,A>AA AAA  ≈ ϑ(ΩΩ 2 2)

<A,A>AA AAA  ≈ ϑ(ΩΩ 2 ω)

<A,A>AA An  ≈ ϑ(ΩΩ n-1 ω)

<A,A>AA An A ≈ ϑ(ΩΩ n )

<A,A>AA AA  ≈ ϑ(ΩΩ ω )

<A,A>AA AA A ≈ ϑ(ΩΩ Ω )

<A,A>(A↑↑5) ≈ ϑ(ΩΩ Ω ω )

<A,A>(A↑↑5)A ≈ ϑ(ΩΩ Ω Ω )

<A,A>A↑↑A ≈ ϑ(εΩ+1)

<A,A>(A↑↑A)A ≈ ϑ(εΩ2)

<A,A>(A↑↑A)(A↑↑A) ≈ ϑ(εΩ22)

<A,A>(A↑↑A)A ≈ ϑ(εΩ2ω)

<A,A>A↑↑AA ≈ ϑ(εΩ2+1)

<A,A>A↑↑AA ≈ ϑ(εΩω)

<A,A>A↑↑AA A  ≈ ϑ(εΩω)

<A,A>A↑↑A↑↑A ≈ ϑ(εε Ω+1 )

<A,A>A↑↑↑A ≈ ϑ(ζΩ+1)

<A,A>A↑↑↑↑A ≈ ϑ(ηΩ+1)

<A,A>A ↑n A ≈ ϑ(φ(n-1,Ω+1))

<A,A>A ↑A A ≈ ϑ(φ(ω,Ω+1)) (or <A,A><A,A>AA)

<A,A><A,A>AAA ≈ ϑ(Ω2)

<A,A><A,A>AA A  ≈ ϑ(Ω2ω)

<A,A><A,A>AA A A ≈ ϑ(Ω2Ω2)

<A,A><A,A>AA AA  ≈ ϑ(Ω2Ω2 ω )

<A,A><A,A>(A↑↑5) ≈ ϑ(Ω2Ω2 Ω2 ω )

<A,A><A,A>A↑↑A ≈ ϑ(εΩ 2+1 )

<A,A><A,A>A ↑n A ≈ ϑ(φ(n-1,Ω2+1))

<A,A><A,A><A,A>AAA ≈ ϑ(Ω3)

<A,A><A,A><A,A><A,A>AAA ≈ ϑ(Ω4)

<A,A><A,A><A,A>...........<A,A><A,A><A,A>AAA with n "<A,A>"s ≈ ϑ(Ωn)

Limit: ϑ(Ωω), the first SGH "catching point".

‾\_(ツ)_/‾

Coming soon: Diagonalizing over this and forming a legion. Goal: ψ(ψ𝐈(0)).