Lately, I've been trying to use ordinals as the argument in FGH and I came out with these:
I don't know if the pattern continues. (EDIT: It does)
If the pattern continues like that, then,
where is the Veblen Function.
The first hierarchy ordinal is
which has the property:
Extending[]
At this point, we need to extend it to create larger ordinals.
- (n f[m](n)'s)
The second hierarchy ordinal is
which has the property:
More[]
Wythagoras and googleaarex extended the notation to easily write . (See comments below)
Let represent rest of expression.
- where m>0.
- where m>0
Then,
There are lots of ordinals that you can create with this and, of course, I wish to extend it more.
Even More[]
Of course, it doesn't end there.
Let's define this:
Then, the third hierarchy ordinal is:
which has the property:
And, of course,[]
We can extend forever!
- where l>1
Using these new rules we can form ordinals like
Also,
The fourth hierarchy ordinal is:
which has the property:
Next[]
is now different from .
New rules:
Let represent a group of n up-arrows.
Then,
And,
Ordinals[]
Close enough :P
THIS IS A WORK IN PROGRESS FOREVER?