03/02/2021, 09:55 PM
Hey Everyone. Haven't been on for a while, been a little busy. I thought I'd post the modified form of the paper.
It details how to construct \( C^\infty \) hyper-operations; minus maybe a few details. But I'm confident I got everything down. In fact, I'm confident that if we do it in the manner Sheldon describes, using a sequence of infinite compositions,
\(
\phi_n(s) = \Omega_{j=1}^\infty \phi_{n-1}(s-j+z)\bullet z\\
\)
Not much really changes. I chose to use the exponential convergents rather than the \( \phi_n \) convergents, simply because it generalizes well to the construction of arbitrary super-functions. All we need really is an exponential convergents. In fact, \( \phi_n \) will look pretty much similarly, because of its exponential nature. The better form of Sheldon's method is that we retain holomorphy. Since I'm only trying to show \( C^\infty \), I figure holomorphy isn't really needed.
I do believe we could definitely use \( \phi_n(s) \) to construct \( e \uparrow^n x \). I believe it's more of an aesthetic issue, and I find it a bit more natural to just use,
\(
\Phi_n(s) = \Omega_{j=1}^\infty e^{s-j}e \uparrow^{n-1} z\bullet z\\
\)
And do away with \( \phi \). Also because this satisfies the more natural equation,
\(
\Phi_n(s+1) = e^s e \uparrow^{n-1} \Phi_n(s)\\
\)
And it removes a bit of the untangling if we were to use \( \phi_n \).
Anyway, here's what I have so far. The proof of \( C^\infty \) hyper-operators is surprisingly copy/paste from the proof of \( C^\infty \) tetration, so long as you pay attention to the generalization, it should be fine.
Any questions, comments or the what have you are greatly appreciated. I spent a lot of time restructuring the paper, but a lot of it is similar to what I wrote before.
Thanks, James
It details how to construct \( C^\infty \) hyper-operations; minus maybe a few details. But I'm confident I got everything down. In fact, I'm confident that if we do it in the manner Sheldon describes, using a sequence of infinite compositions,
\(
\phi_n(s) = \Omega_{j=1}^\infty \phi_{n-1}(s-j+z)\bullet z\\
\)
Not much really changes. I chose to use the exponential convergents rather than the \( \phi_n \) convergents, simply because it generalizes well to the construction of arbitrary super-functions. All we need really is an exponential convergents. In fact, \( \phi_n \) will look pretty much similarly, because of its exponential nature. The better form of Sheldon's method is that we retain holomorphy. Since I'm only trying to show \( C^\infty \), I figure holomorphy isn't really needed.
I do believe we could definitely use \( \phi_n(s) \) to construct \( e \uparrow^n x \). I believe it's more of an aesthetic issue, and I find it a bit more natural to just use,
\(
\Phi_n(s) = \Omega_{j=1}^\infty e^{s-j}e \uparrow^{n-1} z\bullet z\\
\)
And do away with \( \phi \). Also because this satisfies the more natural equation,
\(
\Phi_n(s+1) = e^s e \uparrow^{n-1} \Phi_n(s)\\
\)
And it removes a bit of the untangling if we were to use \( \phi_n \).
Anyway, here's what I have so far. The proof of \( C^\infty \) hyper-operators is surprisingly copy/paste from the proof of \( C^\infty \) tetration, so long as you pay attention to the generalization, it should be fine.
Any questions, comments or the what have you are greatly appreciated. I spent a lot of time restructuring the paper, but a lot of it is similar to what I wrote before.
Thanks, James
