User blog:Nayuta Ito/"Let's make my number bigger" project (Part 1)

Rules:

You can name your function.

One-entry, or something like f(x), must be x+1.

You must not use any operations but plus and minus.

Form of this page:

Most of this page is comparison between my function and FGH. To find out definition easily, for the row of definition, I will write Definition: at the leftmost place. Such as:

Definition:$$f_\omega(x)=f_x(x)$$

Then, let's begin.

The function name is NI, which comes from my name.

Definition:$$NI(x)=x+1$$ Definition:$$NI(x,0)=NI(x-1,x)$$ Definition:$$NI(x,y)=NI(x-1,NI(x,y-1))$$ $$NI(x,c)\simeq f_x(c)$$

K is n zeroes. A is something. Definition:$$NI(Aa,K,x)=NI(Aa-1,x,...x,x)$$ (n+1 times x's) Definition:$$NI(Ax,y)=NI(Ax-1,NI(Ax,y-1))$$

$$NI(1,0,x)=NI(x,x)\simeq f_\omega (x)$$

$$NI(2,0,x)=NI(1,x,x)\simeq f_{\omega 2} (x)$$

$$NI(1,0,0,x)=NI(x,x,x)\simeq f_{\omega^2} (x)$$

$$NI(1,0,0,0,x)\simeq f_{\omega ^3}(x)$$

Definition:$$NI(Aa(1)x)=NI(Aa-1(1)x,x,...,x)$$ (x times x's) Definition:$$NI(Aa,0(1)x)=NI(Aa-1,x(1)x)$$ (x times x's) Definition:$$NI(0(1)A)=NI(A)$$ (x times x's)

$$NI(1(1)x) \simeq f_{\omega^{\omega}} (x)$$

Next: n-dimensional array.