feeding FGH into FGH

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

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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.

Here, I will not define the FGH with transfinite ordinals, and will not prove the limit of it. I'll just write down my idea about it and the possible limit of it. I will keep studying about further definition and proof in the future.

Slow Growing Hierarchy, another hierarchy that grows more slowly, has very unique properties.

for specific function $f(n)$, it looks like that:

$g_{f(\omega)}(n)\approx f(n)$

$g_{f(\omega)}(\omega)\approx f(\omega)$

So,

$\alpha=f_\alpha(\omega)=f_{f_\alpha(\omega)}(\omega)\approx f_{g_\alpha(\omega)}(\omega)$

is actually where the $f_\alpha(\omega)\approx g_\alpha(\omega)$, the point that SGH(n) catches up FGH(n).

There are various results of when it occurs, but most commonly it's $\psi_0(\Omega_{\omega})$.

Thus, $f_{\alpha\mapsto{f_\alpha(\omega)}}(n)\approx f_{\psi_0(\Omega_\omega)}(n)$. It's theoretically the limit of FGH itself. But with Ordinal Collapsing Function that is more powerful and inaccessible ordinals, it is possible for FGH to go beyond its limit.

--D57799 (talk) 06:24, October 5, 2014 (UTC)