04/05/2022, 12:39 PM
Again, thank you for expanding and dumbing down. I'm sure it helps you to pin down the main idea but also helps me to follow.
Sadly I have not enough time to invest right now in order to provide a detailed review of this. If all of this works then we have a mystery: what is the true role of Bennett's family? What it has to do with Goodstein's family? Those are ancient questions... they are rooted on the role of exponentiation and its relation to addition and multiplication. For example once rewritten in group theoretic language, the process of solving superfunction equation, i.e. the engine that makes us able to climb the Goodstein's sequence, presents itself, paired with group conjugation, as analogous to the exp-log concept. That completes a weird analogy that drives my approach since 2012 and that originated in your old post about meta-superfunctions and that we can summarize as follows:
\[\begin{array}
FFunctions&& Numbers\\
f\in G&&z\in\mathbb C\\
group\,\,operation&& multiplication\\
iteration&&exponentiation\\
g^n&&a^n\\
[f,g]&&\log_a(b)\\
conjugation&&exponentation\\
f^g=gfg^{-1}&&a^b\\
Goodstein's\,\,seq.&&tetration\\
S^{H_{n+1}}=H_n&&b^{^nb}={}^{n+1}b\\
\end{array}\]
At first sight the analogy seem to be naive and not able to resist a deeper inspection. But I have good heuristic arguments, involving category theory, that suggest that there is something deep going on here. Also, if you keep in mind that when someone tries to extend the Goodstein equation to negative integers a possible set of solutions is related to the negative ranks of the Bennet hyperoperations, i.e. min-max operations, you can clearly see that there is something mysterious.
Note: Shouldn't the domain \(\mathcal H\) have \(0\) removed?
Sadly I have not enough time to invest right now in order to provide a detailed review of this. If all of this works then we have a mystery: what is the true role of Bennett's family? What it has to do with Goodstein's family? Those are ancient questions... they are rooted on the role of exponentiation and its relation to addition and multiplication. For example once rewritten in group theoretic language, the process of solving superfunction equation, i.e. the engine that makes us able to climb the Goodstein's sequence, presents itself, paired with group conjugation, as analogous to the exp-log concept. That completes a weird analogy that drives my approach since 2012 and that originated in your old post about meta-superfunctions and that we can summarize as follows:
\[\begin{array}
FFunctions&& Numbers\\
f\in G&&z\in\mathbb C\\
group\,\,operation&& multiplication\\
iteration&&exponentiation\\
g^n&&a^n\\
[f,g]&&\log_a(b)\\
conjugation&&exponentation\\
f^g=gfg^{-1}&&a^b\\
Goodstein's\,\,seq.&&tetration\\
S^{H_{n+1}}=H_n&&b^{^nb}={}^{n+1}b\\
\end{array}\]
At first sight the analogy seem to be naive and not able to resist a deeper inspection. But I have good heuristic arguments, involving category theory, that suggest that there is something deep going on here. Also, if you keep in mind that when someone tries to extend the Goodstein equation to negative integers a possible set of solutions is related to the negative ranks of the Bennet hyperoperations, i.e. min-max operations, you can clearly see that there is something mysterious.
Note: Shouldn't the domain \(\mathcal H\) have \(0\) removed?
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)