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There may be some informal 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.

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)

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