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
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 , .
Keeping the integer n big enough ,then
Here we know that
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:
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.