User blog:ArtismScrub/how to destroy a notation 101

Nested factorial notation

a!b = (((...(((a!)!)!)...)!)!)! with b "!"s

this is what i'll be working with here

h
basically, this "extension" is based off how the thing is denoted

n!n is denoted n![2] in the notation, and then the entire thing becomes really abstract and confusing

but what's n!n!? or  n!n! n! ? or  n!n! n! n! n! ... ??

that's easy.

a!↑↑b =  a!a! a! ... a! a!       with b "a!"s

similarly, a! ↑ ↑ ↑b = a! ↑ ↑ a! ↑ ↑ a! ↑ ↑ ... ↑ ↑ a! ↑ ↑ a! ↑ ↑ a! with b "a!"s

or  a! ↑ ↑ ↑ ↑ b = a! ↑ ↑ ↑ a! ↑ ↑ ↑ a! ↑ ↑ ↑ ... ↑ ↑ ↑ a! ↑ ↑ ↑ a! ↑ ↑ ↑ a! with b "a!"s

continue with a!{5}b, a!{6}b, etc.

seems naive, right? of course it is, that's the whole point.

it's not even that powerful--a! {b} c is only comparable to (a!){b+1}c using normal up-arrow notation.

but who cares? we've already exploited the notation beyond the point of any return.

we can already continue with n!{n!}n!, or even something like cg(n!) = n! → n! → n! → ...  → n! → n! → n! with n! "n!"s, which would eventually lead to n!{X}n! where X is a really stupidly huge number.

that's not crazy talk, this is crazy talk
now, i could go the box_m route and define n$ as n!{n}n or something, and then extrapolate with n${n}n or some such trash, but i have a better idea

why not shoehorn this mess into a generalized notation?

no, not CCAN as mentioned earlier. more powerful than that.

BEAF arithmetic.

a! 1 = a!

a!b = a!{a!b-1}a! for b > 1

a!c = a!a!a!...a! a! a! with c "a!"s

continue similarly for a! c, a! c, or higher {a!,b,c,d}.

a![{1}] 1 = a!

a![{1}]b = {a!,b,c,{a!,b-1,1,1,2}}

continue the same way with:

a![{2}]b,  a![{3}]b,  a![{4}]b, ...

a![]b,  a![]b,  a![]b, ...

a!{1}b,  a![{1}]b,  a![[{1}]]b, ...

{a!,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,...)

now we're getting somewhere

we can still continue with:

a! & b! = a*(a-1)*(a-2)*...*3*2*1 & b!, where  a*(a-1)*(a-2)*...*3*2*1 is solved as a dimensional array

continue with a!a! & a!, a! ↑ ↑a! & a!, a! & a! & a!, {a!,a!/2}, {a!,a!/a!}, {{La!,a!}a!,a!&L,a!}a!,a!, who even cares? BEAF isn't even defined properly in the first place

why am i doing this
this is a waste of space but what do i care

i dunno how to conclude this other than... don't take this seriously? this is anything but a good way to extend notations.