FANDOM


...

Basic

f_0,1(n) = f_f_f_..._f_f_n(n)(n)(n)...(n)(n)(n)

f_1,1(n) = f_0,1(f_0,1(...(f_0,1(n))...)) with n levels

f_0,2(n) = a --> f_a,1(n)

f_0,m(n) = a --> f_a.m-1(n)

f_0,0,1(n) = a --> f_0,a(n)

f_0,1,1(n) = a --> f_a,0,1(n)

f_0,0,2(n) = a --> f_0,a,1(n)

etc.

Comparisons

(here values are in fgh (f_@(n)))

0,1 = w+1

1,1 = w+2

0,2 = w*2

0,0,1 = w^2

0,1,1 = w^2+w

0,0,2 = w^2*2

0,0,0,1 = w^3

etc.

So limit is w^w

Introducing the f

f_f(n) = f_0,0,0,...,0,0,1(n) with n entries

f_f+1(n) = f_f(f_f(...(n)...)) with n levels

f_f+0,1(n) = a --> f_f+a(n)

f_f*2(n) = f_f+0,0,0,...,0,0,1(n) with n entries

etc.

f_f_2(n) = f_f_f(f)(n)

f_f_m(n) = f_f_(f_m-1)(f_m-1)(n)

f_f_0,1(n) = a --> f_f_a(n)

f_f_f(n) = f_f_0,0,0,...,0,0,1(n) with n entries

etc.

Comparison

f = w^w

f*2 = w^w*2

f*0,1 = w^(w+1)

f*0,0,1 = w^(w+2)

f^2 = w^(w*2)

Then i got too lazy to continue comparisons.

Please support more if you want.

Extended f

f_f(1,0)(n) = f_f_f_..._f_f_f(n) with n levels

f_f(1,1)(n) = f_f_f_..._f_f_f(1,0)+1(n) with n levels

f_f(0,2)(n) = f_f(f(f(...(f(f(1,0),0),0)...),0),0),0)(n) with n levels

f_f(0,0,1)(n) = f_f(0,f(0,f(0,f(...(0,1)...))))(n) with n levels

etc.

Comparisons

Please support comparisons if you want.

BEYOND!!!

Have any ideas to extend my notation? Put them in the comments!

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