[to do] fully iterative definition of goodstein HOS

[MphLee]

Ya, I always used to get in trouble for that. I would jump too fast. I do most of the math in my head, I only write down things if I have to. I'll give you some examples of what I mean by this notation:

\[
\begin{align}
\alpha \uparrow^0 z &= \alpha \cdot z\\
\alpha \uparrow^{0,\circ m} z &= \alpha^m \cdot z\\
\alpha \uparrow^1 z &= \alpha^z\\
\alpha \uparrow^{1,\circ n} z &= \exp_{\alpha}^{\circ n} z\\
\alpha \uparrow^{0, \circ m} \alpha \uparrow^1 z &= \alpha^m \cdot \alpha^z\\
\alpha \uparrow^{0, \circ m} \alpha \uparrow^{1, \circ n} z &= \alpha^m \cdot \exp_{\alpha}^{\circ n} z\\
\end{align}
\]

I see man! That was pretty clear to me... in fact this kind of notation is not only a notation but the key to unlocking the full algebraic treatment of the topic: that has to do with the major update on my research on ranks theory. What was confusing was how you were using this insider your auxilliary funtions thing... that is what breaks my brain xd the iterated application of partial differintegral xd hahah sorry... that's more a problem on my side...

I can't spoil to much, but it is too exciting. This is not just a notation but is an extension of what I call a Goodstein map into a full monoid homomorphism. This extension is induced naturally by a well known process in group theory (that holds in monoid theory and universal algebra too) and is called an adjunction.

In fact that move is about extending the Goodstein map into a monoid homomorphism from the set of ranks, closed under the "+1" operation, to the free monoid generated by the ranks. This is also where universal algebraic geometry kicks in. I'm hyped by this but I can't spoil... Now that I have one week free I want to write a good introduction to this idea: in that way I can make the terminology crystal clear.

Where my point was, when we use regular iteration (the schroder iteration/the fractional calculus iteration), we get your rule:

\[
\alpha \uparrow^{n,\circ y} \alpha \uparrow^{n+1} z = \alpha \uparrow^{n+1} z+y\\
\]
But for god's sake! Do we need all these uparrows? Let's get rid of one:
\[
\alpha \uparrow^{(n,\circ y),n+1} z=\alpha \uparrow^{n+1} z+y\\
\]
So there's a correspondence between this notation and a translation algebra.
So I write:
\[
\alpha \uparrow^{(k_1,\circ m_1),...,(k_n,m_n)} z = \alpha \uparrow^{(k_1,\circ m_1)}\cdots \alpha \uparrow^{(k_n,\circ m_n)} z\\
\]
Which just saves time because it gets rid of \(n-1\) up arrows and I'm not constantly retyping that out, lol. It's not the prettiest, but it gets the job done.

Exactly, this is all implicit in my Goodstein maps yoga. What you write with your second last expression I write as

\[{\bf h}^y(n){\bf h}^1(n^+)={\bf h}^1(n^+){\bf h}^y(0)\]

That is all there is about my equivariant upgrade of Goodstein. But, if we expand the ranks \(n\in J\), where \(J\) is a discrete dynamical system, to the free monoid \({\sf F}(J)\) generated on the set of ranks as symbols, we have that a word \(w\in {\sf F}(J)\) is, by definition of free monoid, is just a finite sequence in the \(J\)-symbols \(w=\prod_{i=0}^d (j_i,n_i)\) where \(j_i\in J\) and where \((j,m)(j,n)=(j,n+m)\). By general abstract nonsensense we have an universal extension of the Goodstein map \({\bf h}:J\to {\rm End}(X)\) to a monoid homomorphism \({\bf h}^\dagger:{\sf F}(J)\to {\rm End}(X)\) that satisfies

\[{\bf h}^\dagger(w)={\bf h}^\dagger(\prod_{i=0}^d (j_i,n_i))=\Omega_{i=0}^d{\bf h}^\dagger((j_i,n_i))=\Omega_{i=0}^d{\bf h}(j_i)^{\circ n_i}\]


Where the omega is to be undesrtood as an abstract iterated composition of functions, that's why I omit the bullett: \(\Omega_{i=0}^d{\bf h}(j_i)^{\circ n_i}={\bf h}(j_0)^{\circ n_0}\circ {\bf h}(j_1)^{\circ n_1}\circ ...\circ {\bf h}(j_d)^{\circ n_d}\). Clearly we have \(\alpha \uparrow^{(n,\circ y),n+1} z=\alpha \uparrow^{n+1} z+y\) but I write at the function composition level as

\[{\bf h}^\dagger ((n,y)(n^+,1))={\bf h}^\dagger ((n^+,1)(0,y))\]

Here if you are comfortable with group presentations, ring theory, ideals and coordinate rings in AG (Algebraic Geometry=Alexander Grothendieck  ), you can see where this is going.... DDDD Notice that the elegant and minimal expression is given by choosing the successor to be the seed and the seed to coincide with the zerot-th rank... note that in you construction the seed is still the successor but it is not the zero rank anymore.

The point is: what is not implicit in my yoga of Goodstein maps is how to extend to the monoid freely genereated by the ranks to something that allows \(y\in \mathbb R\) in the expression \(\uparrow^(j,y)\)... and your construction gives a model of the extended theory.

Additionally, if you were confused; \(s\Big{|}_{\mathbb{N}}\) means the variable \(\Re(s) >0\) is now restricted to \(s \in \mathbb{N}\)--which is standard notation, though not very popular. I like it though.

EDIT:

I really think the fractional calculus is the approach for you though. It works off of \(\mathbb{N}\), but constructs it for \(T = \mathbb{R}^+\). Which is to say, your PEANO idea, is totally satisfied by the bounded analytic hyper-operators.

Yea thanks you xD... that was one of the points that confused me.

I apologize, as confused you are by my work; I can empathize with the feeling I have for some of your work  

But I think the bounded analytic hyper-operators are at least an example of "something that looks like this monoid/\(\mathbb{N}\)/goodstein thing you have". But maybe I'm misinterpreting.... I think this is super important because; everything in this scenario relates to Mellin transforms and Generating functions. And you know what's next door to the mellin transform... the fourier transform. What's next door to that is Hilbert spaces. Next thing you know we're doing categorical fourier transforms, lmao (Which are a thing!).

In some sense you are right. But the material in this thread is not really my work but just some speculations... I'm working the foundations for tackling the problems I'm describing in this thread.
It is not your fault if you don't get the terminology right... I need to do a good work explaining all of this from zero... I'll do this. It is so much easier that your differintegral/transform stuff...
It is like the difference between the Cappella Sisitina (your work) and something like the Giza pyramids (my work).

---

Notice that all of this algebra of ranks, and the remark about translation algebra, was foreseen by you back in August 2013 and we probably came at this almost independently (see a private message I sent you back in June 2013 ) before our research diverged. It is hard to tell who came there first since I was thinking about the algebra of ranks almost since the beginning of my journey... back in summer 2010, but I was deeply influenced by this November 2011 thread by you... at the point that it made me see how that idea I had back in 2010 could work... I really mean this... that thread made me have a clear grand vision... something that grew  so big that in 2013 I felt the need to join the forum and send a PM to you. We can say that the last 12 years were just about getting the right tools to develop that vision.

The map \(n\mapsto [n]\) is just the prototype of Goodstein map I'm talking about, \(h(n)=[n]\)! And the free monoid nonsense is about making satisfy \([(s,n)(t,m)]=[(s,n)][(s,m)]\) where \([(s,n)]=[s]^{\circ n}\). Then we have \(T^m[s]=[s][0]^{\circ m}\) and clearly \([s][s+1]=T[s+1]=[s+1][0]\). All that Goostein theory is about studying in full generality maps \(h:J\to M\) that behaves as \([-]:\mathbb N\to (\mathbb N\times \mathbb N \to\mathbb N)\).