User blog:ArtismScrub/Hyper-Alpha Notation (WIP?)

Hyper-Alpha Notation is a notation I came up with at school today that reaches ε0. The notation is based on mapping symbolic recursion to hyper-operators.

(ε0 is big news for me because if you'll recall, I've made extensions to ωω-level notations before that only reached ωω ω, so reaching ε0 with an original notation feels like kind of a big deal.)

The notation is defined as follows:

BASE CASE (no operators):  = a×b

TERMINATION CASE: ★A = a

RECURSIVE CASE: ★A>★ where ★ is any string of operators

EXPRESSIVE CASE: ★☐c = ★☐☐☐☐...☐☐☐☐ with c "☐"s

CONCATENATION CASE: ★☐★A = ★☐★☐★☐★...☐★☐★☐★ with c "☐★"s

DIAGONALIZATION CASE: ★☐A = ★☐b

Of course, putting this recursively may seem rather abstract, so I'll give some progression.

Ac = a ↑c b, this should be plain to see through the first 3 rules.

AA = Ab = a ↑b a, or roughly ω in FGH.

AAA iterates AA and so on...

AAAc = {a,b,c,2} in BEAF, or roughly ω+c in FGH. This notation does indeed start off with exact equivalence to BEAF. This was not intentional, but the notation must begin somewhere.

AAAA = <a,a>AAAb = {a,a,b,2} in BEAF, or roughly ω2 in FGH.

<a,b>AAAAAc = {a,b,c,3} in BEAF.

Generally, <a,b>AAAAAA...AAAAAc with d "AA"s = {a,b,c,d+1} in BEAF, or roughly ω(d+1)+c in FGH.

<a,b>AAA = <a,a>AAAAAA ... AAAAAA with b "AA"s = {a,a,a,b} in BEAF, or roughly ω2 in FGH.

<a,b>AAAAc = {a,b,c,1,2} in BEAF or about ω2+c

<a,b>AAAAA = {a,a,1,2,2} in BEAF or about ω2+ω

Essentially, if <a,b>★ = {a,b,c,d} then <a,b>AAA★ = {a,b,c,d,2}

This continues with:

<a,b>AAAAAA★ = {a,b,c,d,3}

<a,b>AAAAAAAAA★ = {a,b,c,d,4}

And generally:

<a,b>AAAAAAAAA...AAAAAAAAA★ = {a,b,c,d,e+1} or about ω2(e+1)+ωd+c

Continue with:

<a,b>AAAA = {a,a,a,a,b} ≈ ω3

<a,b>AAAAA = {a,a,a,a,a,b} ≈ ω4

<a,b>AA c  = {a,a,a,a,...,a,a,a,a,b} with c "a"s ≈ ωc

And then we have <a,b>AA A, which entirely catches up with linear BEAF, being equal to {a,b+1(1)2} and comparable to ωω in FGH. And that's just the third exponent!

For <a,b>AA A A and beyond, the BEAF equivalence is over. <a,b>AA A A is comparable to ωω+1.

Continuing:

<a,b>AA A AA ≈ ωω+ω

<a,b>AA A AAA ≈ ωω+ω2

<a,b>AA A AA A  ≈ ωω2

<a,b>AA AA  ≈ ωω+1

<a,b>AA AAb  ≈ ωω+b

<a,b>AA AAA  ≈ ωω2

<a,b>AA AA  ≈ ωω 2

<a,b>AA AAA  ≈ ωω 3

<a,b>AA A c  ≈ ωω c

<a,b>AA A A  ≈ ωω ω --remember, this is the place I kept getting stuck on, and I've already surpassed it!

<a,b>AA A A A  ≈ ωω ω +1

<a,b>AA A A A  ≈ ωω ω+1

<a,b>AA A AA  ≈ ωω ω 2

<a,b>AA A AAA  ≈ ωω ω 3

<a,b>AA A A c   ≈ ωω ω c

<a,b>AA A A A   ≈ ωω ω ω

etc.

<a,b>A↑↑c ≈ ω↑↑(c-1)

So the limit of this notation so far is <a,b>A↑↑A ≈ ε0, the big hurdle.

This is most certainly my first time managing to reach such recursive heights. I hope to extend this notation, so I can soon be able to define beyond A↑↑A, A↑↑↑A, A ↑A A, etc., or even continue by iterating <A,A>★ and reach what Bowers hoped to achieve with his array-arrays and "legions".