04/08/2008, 09:04 PM
OK, I think we need to look at all the possibilities before we go too far into determining notation.
We seem to be stuck on two things that are very closely related: auxiliary hyper-operations and iterated hyper-operations, which are practically the same things, but from two different viewpoints (an example difference is that iterated hyper-3 is auxiliary hyper-4). Since the term "auxiliary" is new and "iterated" is old and venerable, it is more appropriate to call them "iterated" hyper-operations, although either term could suffice. A comparison between the notation I used and the notation that GFR used:
} = \mathtt{b[N]<h>x})
however, since GFR's notation requires angle-brackets around the 'y', it prevents it from being used with slash-notation, especially for some inverse hyperops. To illustrate the difficulties, I will use GFR's instead.
} <br />
& \mathtt{b[N]<h>x}<br />
& \text{iterated hyper-operations}<br />
\\<br />
b =<br />
& \mathtt{z/[N]\^h(x)} <br />
& \mathtt{z/[N]<h>x}<br />
& \text{auxiliary hyper-roots}<br />
\\<br />
h =<br />
& \mathtt{b[N]\^{\backslash}z(x)} <br />
& \mathtt{b[N]< >x{\backslash}z}?<br />
& \text{auxiliary hyper-logarithms}<br />
\\<br />
x =<br />
& \mathtt{b[N]\^(-h)(z)} <br />
& \mathtt{b[N]<-h>z}<br />
& \text{negatively iterated hyper-operations}<br />
\end{tabular}<br />
)
Fortunately, however, we do not need a notation for auxiliary hyper-logarithms, because:
} <br />
= \left({}^N_b\begin{tabular}{|c} z \\\hline\end{tabular}\right)<br />
- \left({}^N_b\begin{tabular}{|c} x \\\hline\end{tabular}\right)<br />
)
So if neccessary, this can be written
which means we really don't need either my notation, nor GFR's notation for auxiliary hyper-logarithms. Also, as you can see, we also don't need a notation for the auxiliary inverse, because these can be represented by negatively iterated hyper-operations. What this means is that we only need a notation for auxiliary hyper-roots.
Andrew Robbins
We seem to be stuck on two things that are very closely related: auxiliary hyper-operations and iterated hyper-operations, which are practically the same things, but from two different viewpoints (an example difference is that iterated hyper-3 is auxiliary hyper-4). Since the term "auxiliary" is new and "iterated" is old and venerable, it is more appropriate to call them "iterated" hyper-operations, although either term could suffice. A comparison between the notation I used and the notation that GFR used:
Fortunately, however, we do not need a notation for auxiliary hyper-logarithms, because:
So if neccessary, this can be written
Andrew Robbins