Hey everyone, I've been interested in large numbers for a long time and I am really excited to discover this wiki for the first time. After I saw all these mind-blowing fast-growing functions on your site I just HAD to create my own! :D So would you mind having a look at my notation? It's called Super-Lambda Array Notation, a new array notation I invented that I think goes beyond the power of ExE, BEAF, and HAN. I'm kinda new at this googology stuff, so feedback is welcome :P In making this notation I tried to start from the ground up and find ways to strengthen the notation on every level. The notation has the following rules:
- Every expression must start with Λ (capital Greek lambda).
- Expressions can contain any list of numbers.
- Λn = n^n
- Λ#,1 = Λ# where # denotes rest of array.
- Λn,# = Λ(Λn),#-1
- Λa,b,# = Λ(Λ(...(b times)...Λ(a,Λ(a,Λ(a,Λ(a,...)...)...(b times)...)...#)...#)...#)
This is only the first level of SLAN, but already it is powerful than 2-entry arrays in ExE, BEAF, and HAN. Notice how in the last rule the list gets recursed over in both arguments. This results in amazingly, catastrophically larger numbers than any other 2-argument notation I've seen! I think the growth rate is f_w but I'm not too good at the FGH stuff, I'll leave it for other people to prove. Another distinguishing property of Super-Lambda Array Notation concerns the use of Greek letters instead of ordinary English and ASCII letters. Anyways, we can move on to the extension, Extended Super-Lambda Array Notation (ESLAN):
- At some point in the expression we can have a jump-marker Γ (Greek capital gamma).
- Let @ represent the expression up to the jump-marker, and # is the rest of the expression.
- Λ1Γ# = 1
- ΛnΓ# = Λn-1Γ(Λn-1Γ(Λn-1Γ(...(b times)...Λn-1#)))
- Λa,b,@Γ# = Λ(@Λ(@...(b times)...Λ(@a,Λ(@a,Λ(@a,Λ(@a,...)...)...(b times)...)...#)...#)...#)
With only five simple rules I think this extends Super-Lambda Array Notation to reach a strength level of f_w^2! To extend to higher-level notations we need to introduce more dimensional notations, which brings us to Hyper-Dimensional Mega-Extended Super-Lambda Array Notation (whew, that's a long name :P):
- We use Γ2 for a double jump-marker, which means a second-dimensional break.
- Γ3 is the triple jump-marker, and so on. ΓΓ is the jump-jump marker, Γ^3 the jump-jump-jump marker or jump-3 marker, and so on to jump^jump markers, jump^^^jump markers, Λ(jump,jump) markers, ........ We call these all jump-structures.
- The accumulator entry in an X-tuple jump-structure is the X-1th entry. The accumulator in a jump-jump marker is the bth entry (when solving rule 5 above), for a jump-jump-jump marker it's the Λb'th entry, and so on for other jump-structure markers.
- The three simple rules are:
- Λn = n^n
- Λ1Δ@ = Λ@ where Δ is any jump-structure-marker
- Λ#Δ@ = Λ#Δ(Λ#Δ(Λ#Δ@(Λ#Δ...))) (b times, where b is the value of the accumulator entry and Δ is a jump-structure marker)
And of course we can define all sorts of numbers. I define Dave's Number as ΛgoogolΔgoogol where Δ is a ΛjumpΔjump marker where Δ is a ΛjumpΔjump marker where Δ is a ΛjumpΔjump marker where ... a total of googol times! Yikes!!!!! I think this notation is SO strong that it might be beyond any known ordinal notation... I have NO idea what it could be in FGH! I'm wondering if you have any idea what it would be. Still a work in progress, and there may be some things that aren't well-defined, but I hope you can sort this all out :P Maybe someone can make an article about this? I think it's quite a novel discovery in googology and might deserve an article in the wiki! Thanks in advance!