User blog:Ubersketch/Hypernomial Hierarchy

Hypernomial Hierarchy
F(n) = 0

S(n) = n+1

FS(n,m) = mth member of n's fundamental sequence

The fundamental sequence of a limit hypernomial f(w) = FS(f(w),f(n))

w is the hypernomial with fundamental sequence FS(w, n)

f(n) is any function


 * 1) is any string (can be empty)

1. f(a)( [0]b ) = f(a)

2. f(a)( [S(b)]c ) = S(f(a)( [b]c ))

3. f(a)( #b# ) = FS(b,a) iff b is a limit hypernomial

4. f(a)( #{b}# ) = limit hypernomial with fundamental sequence FS(f(a)( #{b}# ),a) iff b is a limit hypernomial

5. f(a)([#[b]c) = g(a)([b]c) where g(a) = f(a)(#)

We will call this notation h(f(S)) where S is the set of allowed hypernomials and f(n) is the only function allowed to start the string.

For our definitions the only hypernomials we are allowed to use, H, within the notation h(F(n)) are defined as so:

0

w

S(H) where a is a member of H

h(F(H))

H is the hypernomial hierarchy.

Done, that was pretty simple.

Explanation
Now this may be pretty confusing to ya'll but bear with me.

First the notation. It's a function based off of the Slow Growing Hierarchy. In essence we are allowed to iterate the Slow Growing Hierarchy and iterate the resulting function and so on...

Second, hypernomials. Hypernomials are like ordinals, but they aren't since hypernomials with different fundamental sequences are not the same hypernomial.

Thirdly, the hierarchy itself. H is made up of repeated applications of hypernomials on this system starting with 0 and w with the successor function and the notation itself using only hypernomials defined in the H itself.

Much better.