Zeration

[JmsNxn]

I want to point to this other thread where a novel possible definition for operation preceding addition is proposed by user Danteman163.
It seems to me that this idea was never been described before.

https://math.eretrandre.org/tetrationfor...32#pid9332

Hmm I actually really like that. Especially his description. Is that a viable operator though...? Personally I still like the \(\Gamma\) function interpretation.

\[
a \langle 0 \rangle b = a + b\\
\]

Then:

\[
\begin{align}
a \langle -1 \rangle b &= a +1\\
a \langle -2 \rangle b &= a+1\\
a \langle -3 \rangle b &= a+1\\
&\vdots
\end{align}
\]

But, there's singular behaviour at each of these points. And that small perturbations in the hyper operational indexes are wild and destructive. Additionally, the action of going back becomes idempotent. There isn't a 1-1 in the mapping \(\langle q \rangle \to \langle q-1\rangle\). But this only happens at the singular points at \(-1,-2,-3,...\). You need local data near \(q \approx -2\) to pull back to addition again. Despite \(q =-2\) alone making \(q=-1\) addition when you try to naively map forward \(\langle q \rangle \to \langle q+1\rangle\). Where doing this naively just iterates the predecessor. Doing it more difficultly requires iterating a neighborhood of the predecessor.

I apologize. I like this idea from zeration. But honestly. It should be successorship all the way down, but at each singularity, there is different behaviour in a neighborhood of each successorship.