Notations and Opinions

[andydude]

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:

\( \mathtt{b[N]\^h(x)} = \mathtt{b[N]<h>x} \)

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.

\(
\begin{tabular}{c|l|l|l}
\text{inv} & \text{my} & \text{GFR's} & \text{name} \\
\hline
{z =}
& \mathtt{b[N]\^h(x)}
& \mathtt{b[N]<h>x}
& \text{iterated hyper-operations}
\\
b =
& \mathtt{z/[N]\^h(x)}
& \mathtt{z/[N]<h>x}
& \text{auxiliary hyper-roots}
\\
h =
& \mathtt{b[N]\^{\backslash}z(x)}
& \mathtt{b[N]< >x{\backslash}z}?
& \text{auxiliary hyper-logarithms}
\\
x =
& \mathtt{b[N]\^(-h)(z)}
& \mathtt{b[N]<-h>z}
& \text{negatively iterated hyper-operations}
\end{tabular}
\)

Fortunately, however, we do not need a notation for auxiliary hyper-logarithms, because:

\( h
= \mathtt{b[N]\^{\backslash}z(x)}
= \left({}^N_b\begin{tabular}{|c} z \\\hline\end{tabular}\right)
- \left({}^N_b\begin{tabular}{|c} x \\\hline\end{tabular}\right)
\)

So if neccessary, this can be written \( h = \mathtt{b[N]{\backslash}z - b[N]{\backslash}x} \) 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