User blog:D57799/feeding FGH into FGH

There may be some imformal or invalid step in my proof. Just point it out. There's still a lot for me to learn.

Feeding FGH into itself was considered before. I just want to find the limit of it and probably some usage of it.

It looks like this:

1. $$f_{f_\alpha(n)}(n)$$, or

2. $$f_{f_\alpha(\omega)}(n)$$

1. and 2. is very different. 1. comes to an limit very early, while 2. can go very far.

The first situation
the limit of $$f_{f_\alpha(n)}(n)$$ is $$f_{f_{f_{...}(n)}(n)}(n)$$, while n is an integer.

when m,n are integers, for integer n large enough, $$2\uparrow^{m}n>f_m(n)>2\uparrow^{m-1}n$$.

Therefore, $$2\uparrow^{2\uparrow^{2\uparrow^{...}n}n}n>f_{f_{f_{...}(n)}(n)}(n)>2\uparrow^{2\uparrow^{2\uparrow^{...}n-1}n-1}n$$

Keeping the integer n big enough ,then $$f_{\omega+1}(n)>{\{n,n,1,2}\}=n\uparrow^{n\uparrow^{n\uparrow^{...}n}n}n>2\uparrow^{2\uparrow^{2\uparrow^{...}n}n}n$$

Here we know that $$f_{\omega+1}(n)>f_{f_{f_{...}(n)}(n)}(n)>2\uparrow^nn>f_\omega(n)$$

So, $$f_{\omega+1}(n)>f_{\alpha\mapsto{f_\alpha(n)}}(n)>f_\omega(n)$$

The second situation
The second situation is more fierce and powerful. Since it involves FGH with n replaced by ordinals. It will be harder to define.

To be continued...