User:Vel!/pu/Cookiefonster's alphabet hierarchy

Part 1: simple alphabet and & notation
Start with a0(n) = 10n and a(m+1)(n) = a(m)n(n)

Then aA(n) = an(n)

aA1, aA2, aAA, aAAA, ...

aB -> aAA...AA

aC -> aBB...BB

...

aZ -> aYY...YY

Then A = &[1], B = &[2], C = &[3], ...

Then &[A] -> &[n] - that's &[&[1]].

Then &[&[2]], &[&[A]] = &[&[&[1]]], ...

Formal definition

 * $ is anything of the form &[...]
 * @ can be anything
 * Main rules
 * Rule M1: a0(n) = 10^n
 * Rule M2: a@0(n) = a@(n)
 * Rule M3: a(m+1)(n) = a(m)n(n)
 * Rule M4: a@$(n) = a@$[n](n)
 * Fundamental sequences
 * Rule F1: &[1][n] = n
 * Rule F2: If m > 1 or @ is not nothing,
 * Rule F2A: If n = 1, &[@m][1] = &[@m-1]
 * Rule F2B: If n > 1, &[@m][n] = &[@m][n-1]&[@m-1]
 * Rule F3: &[@0] = &[@]
 * Rule F4: Otherwise, &[@$][n] = &[@$[n]]

Googol regiment

 * Googol = a0(100)
 * Googolplex = a0(a0(100))
 * Googolduplex = a0(a0(a0(100)))
 * Googoltriplex = a0(a0(a0(a0(100))))
 * Googolquadriplex = a(0)5(100)
 * Googolquinplex = a(0)6(100)
 * Googolsexplex = a(0)7(100)
 * Googolseptiplex = a(0)8(100)
 * Googoloctiplex = a(0)9(100)
 * Googolnoniplex = a(0)10(100)
 * Googoldeciplex = a(0)11(100)

Iteroogol regiment

 * Iteroogol = a1(100)
 * Iteroogolplex = a0(a1(100))
 * Iteroogolduplex = a0(a0(a1(100)))
 * Iteroogoltriplex = a0(a0(a0(a1(100))))
 * Iteroogolquadriplex = a0(a0(a0(a0(a1(100)))))

Now...


 * Iterduoogol (or iteroogolpliterex) = a1(a1(100))
 * Iterduoogolplex = a0(a1(a1(100)))
 * Itertrioogol (or iteroogoldupliterex) = a1(a1(a1(100)))
 * Iterquadroogol (or iteroogoltripliterex) = a1(a1(a1(a1(100))))
 * Iterquintoogol = a(1)5(100)
 * Itersextoogol = a(1)6(100)
 * Iterseptoogol = a(1)7(100)
 * Iteroctoogol = a(1)8(100)
 * Iternonoogol = a(1)9(100)
 * Iterdecoogol = a(1)10(100)
 * Iteroogoldecipliterex = a(1)11(100)

But that's not the end!


 * Googolpliterex = a1(a0(100))
 * Googolplexipliterex = a1(a0(a0(100)))
 * Googolplexipliterexplex = a0(a1(a0(a0(100))))

And we might go down a step and have


 * Googolpliterexiplex = a0(a1(a0(100)))
 * Googolpliterexiduplex = a0(a0(a1(a0(100))))

Meanwhile...


 * Iteroogolplexipliterex = a1(a0(a1(100)))
 * Iteroogolduplexipliterex = a1(a0(a0(a1(100))))
 * Iterduoogolplexipliterex = a1(a0(a1(a1(100))))
 * Iteroogolplexidupliterex = a1(a1(a0(a1(100))))

Endless possibilities...