User blog:ArtismScrub/Hyper-Alpha Notation Part II: Arrows to Alphas (STILL WIP)

See Hyper-Alpha Notation. Last time, I had defined up to A↑↑A, which reached strength of ε0. Now, I've found a simple method to generalize this towards pentation, hexation, etc. (I actually came up with this some time before I even made the original blog post, but I hadn't bothered to test it to its limits.)

Consider the properties:

An = AAAAAA........AAAAAA with n "A"s

A★A = (A★)A

UPGRADE:

A ↑a+1 b = A ↑a A ↑a A ↑a ... ↑a A ↑a A ↑a A with b "A"s

A ↑n ★A = (A ↑n ★) ↑n A

This simple change will cover us for any number of up-arrows.

Of course, like last time, I'll give some recursive progression.

A↑↑A ≈ ε0 (this was where we left off last time)

(A↑↑A)A ≈ ε0+1

(A↑↑A)An ≈ ε0+n

(A↑↑A)AA ≈ ε0+ω

(A↑↑A)(A↑↑n) ≈ ε0+(ω↑↑(n-1))

(A↑↑A)(A↑↑A) ≈ ε02

(A↑↑A)n ≈ ε0n

(A↑↑A)A ≈ ε0ω = ωε0+1

(A↑↑A)AA ≈ ε0(ω2)

(A↑↑A)A A  ≈ ε0(ωω)

(A↑↑A)(A↑↑n) ≈ ε0(ω↑↑n)

(A↑↑A)(A↑↑A) ≈ ε02

(A↑↑A)(A↑↑A)A ≈ ε02ω

(A↑↑A)(A↑↑A)(A↑↑A) ≈ ε03

(A↑↑A)(A↑↑A) A  ≈ ε0ω

<a,b>(A↑↑A)(A↑↑A) A A  ≈ ε0ω ω

<a,b>(A↑↑A)(A↑↑A) (A↑↑n)  ≈ ε0(ω↑↑n)

<a,b>(A↑↑A)(A↑↑A) (A↑↑A)  ≈ ε0ε0

<a,b>(A↑↑A)(A↑↑A) (A↑↑A) (A↑↑A)  ≈ ε0ε0 ε 0

<a,b>(A↑↑A)↑↑n ≈ ε0↑↑(n-1)

<a,b>(A↑↑A)↑↑A = <a,b>A↑↑AA ≈ ε1

<a,b>(A↑↑AA)A ≈ ε1ω

<a,b>(A↑↑AA)(A↑↑A) ≈ ε1×ε0

<a,b>(A↑↑AA)(A↑↑AA) ≈ ε12

<a,b>(A↑↑AA)(A↑↑AA) A  ≈ ε1ω

<a,b>(A↑↑AA)(A↑↑AA) (A↑↑A)  ≈ ε1ε0

<a,b>(A↑↑AA)(A↑↑AA) (A↑↑AA)  ≈ ε1ε1

<a,b>(A↑↑AA)(A↑↑AA) (A↑↑AA) (A↑↑AA)  ≈ ε1ε1 ε 1

<a,b>A↑↑AAA ≈ ε2

<a,b>(A↑↑AAA)(A↑↑AAA) ≈ ε22

<a,b>(A↑↑AAA)(A↑↑AAA) (A↑↑AAA)  ≈ ε2ε2

<a,b>A↑↑AAAA ≈ ε3

<a,b>A↑↑An ≈ εn-1

<a,b>A↑↑AA ≈ εω

<a,b>A↑↑(AAA) ≈ εω+1

<a,b>A↑↑(AAAA) ≈ εω2

<a,b>A↑↑(AAA) ≈ εω2

<a,b>A↑↑(AA A ) ≈ εωω

<a,b>A↑↑(A↑↑n) ≈ εω↑↑n

<a,b>A↑↑(A↑↑A) ≈ εε 0

<a,b>A↑↑(A↑↑(A↑↑A)) ≈ εε ε 0

<a,b>A↑↑↑A ≈ ζ0

<a,b>(A↑↑↑A)A ≈ ζ0ω

<a,b>(A↑↑↑A)(A↑↑A) ≈ ζ0×ε0

<a,b>(A↑↑↑A)(A↑↑↑A) ≈ ζ02

<a,b>(A↑↑↑A)(A↑↑↑A) A  ≈ ζ0ω

<a,b>(A↑↑↑A)(A↑↑↑A) (A↑↑A)  ≈ ζ0ε0

<a,b>(A↑↑↑A)(A↑↑↑A) (A↑↑↑A)  ≈ ζ0ζ0

<a,b>(A↑↑↑A)(A↑↑↑A) (A↑↑↑A) (A↑↑↑A)  ≈ ζ0ζ0 ζ 0

<a,b>(A↑↑↑A)↑↑A ≈ εζ 0+1

<a,b>(A↑↑↑A)↑↑AA ≈ εζ 0+2

<a,b>(A↑↑↑A)↑↑AA ≈ εζ 0+ω

<a,b>(A↑↑↑A)↑↑(A↑↑A) ≈ εζ 0+ε0

<a,b>(A↑↑↑A)↑↑(A↑↑AA) ≈ εζ 0+ε1

<a,b>(A↑↑↑A)↑↑(A↑↑AA) ≈ εζ 0+εω

<a,b>(A↑↑↑A)↑↑(A↑↑(A↑↑A)) ≈ εζ 0+εε 0

<a,b>(A↑↑↑A)↑↑(A↑↑↑A) ≈ εζ 02

<a,b>(A↑↑↑A)↑↑(A↑↑↑A)A ≈ εζ 0ω

<a,b>(A↑↑↑A)↑↑(A↑↑↑A)(A↑↑↑A) ≈ εζ 02

<a,b>(A↑↑↑A)↑↑(A↑↑↑A)(A↑↑↑A) (A↑↑↑A) ≈ εζ 0ζ0

<a,b>(A↑↑↑A)↑↑(A↑↑↑A)↑↑A ≈ εε ζ 0+1

<a,b>(A↑↑↑A)↑↑(A↑↑↑A)↑↑(A↑↑↑A) ≈ εε ζ 02

<a,b>A↑↑↑AA ≈ ζ1

<a,b>(A↑↑↑AA)↑↑A ≈ εζ 1+1

<a,b>(A↑↑↑AA)↑↑(A↑↑A) ≈ εζ 1+ε0

<a,b>(A↑↑↑AA)↑↑(A↑↑↑A) ≈ εζ 1+ζ0

<a,b>(A↑↑↑AA)↑↑(A↑↑↑AA) ≈ εζ 12

<a,b>(A↑↑↑AA)↑↑(A↑↑↑AA)↑↑(A↑↑↑AA) ≈ εε ζ 12

<a,b>A↑↑↑AAA ≈ ζ2

<a,b>A↑↑↑AA ≈ ζω

<a,b>A↑↑↑(A↑↑A) ≈ ζε 0

<a,b>A↑↑↑(A↑↑↑A) ≈ ζζ 0

<a,b>A↑↑↑(A↑↑↑(A↑↑↑A)) ≈ ζζ ζ 0

<a,b>A↑↑↑↑A ≈ η0

<a,b>(A↑↑↑↑A)A ≈ η0ω

<a,b>(A↑↑↑↑A)↑↑A ≈ εη 0+1

<a,b>(A↑↑↑↑A)↑↑↑A ≈ ζη 0+1

<a,b>A↑↑↑↑AA ≈ η1

<a,b>A↑↑↑↑AA ≈ ηω

<a,b>A↑↑↑↑(A↑↑A) ≈ ηε 0

<a,b>A↑↑↑↑(A↑↑↑A) ≈ ηζ 0

<a,b>A↑↑↑↑(A↑↑↑↑A) ≈ ηη 0

<a,b>A↑↑↑↑(A↑↑↑↑(A↑↑↑↑A)) ≈ ηη η 0

<a,b>A↑↑↑↑↑A ≈ φ(4,0)

<a,b>A↑↑↑↑↑↑A ≈ φ(5,0)

<a,b>A ↑n A ≈ φ(n-1,0)

<a,b>A ↑A A ≈ φ(ω,0)

<a,b>A ↑AA A and beyond: How to work with these? Only God knows--but they must form some massive alpha strings--and utterly unspeakable numbers when solved.

Nah, just kidding, I haven't generalized that far yet!

Just the previous Thursday, I was shocked to reach ε0. I hadn't expected to reach such recursive heights, and now I've made it FAR further than that.

This is certainly a milestone. How many people have even managed φ(ω,0) by themselves? Probably a minority out of all.

Again, I do hope to generalize this further for higher A ↑★ A, and of course, iterate <A,A>★ and form a fully functional legiattic notation, reaching ψ(Ωω) and beyond. This will either be a simple modification or a distant dream.