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?**