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. , or

2.

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 is , while n is an integer.

when m,n are integers, for integer n **large enough** , .

Therefore,

Keeping the integer n big enough ,then

Here we know that

So,

## 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 , it looks like that:

So,

is actually where the , the point that SGH(n) catches up FGH(n).

There are various results of when it occurs, but most commonly it's .

Thus, . 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.