Circulation and the Fast-Growing Hierarchy

Circulation and the Fast-Growing Hierarchy - Printable Version

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Circulation and the Fast-Growing Hierarchy - Catullus - 06/04/2022

For an ordinal alpha greater than one, $[Image: svg.image?f_\alpha(x)]$ in the Fast-growing hierarchy is greater than 2[alpha+1]x. So should not the super function of 2 circulated to the x be less than $[Image: svg.image?f_\omega(x)]$ ? But on the Googology Wiki page for circulation says that "Using only nonnegative integers, there are only four cases where circulation is defined:".

RE: Circulation and the Fast-Growing Hierarchy - JmsNxn - 06/05/2022

(06/04/2022, 03:17 AM)Catullus Wrote: For an ordinal alpha greater than one, f $[Image: svg.image?\alpha]$ (x) in the Fast-growing hierarchy is greater than or equal to 2 ( $[Image: svg.image?\alpha]$ +1)-ated to the x. So should not the super function of 2 circulated to the x be less than f $[Image: svg.image?\omega]$ (x)? But on the Googology Wiki page for circulation says that "Using only nonnegative integers, there are only four cases where circulation is defined:".

Could you post a link to what a circulation is? Sorry, I'm unaware of that. Fast growing hierarchy's usually don't get much attention on this forum, it might be better to post it on a forum more geared towards that. I'm interested in what exactly this means though.

Regards, James

RE: Circulation and the Fast-Growing Hierarchy - Catullus - 06/06/2022

(06/05/2022, 11:38 PM)JmsNxn Wrote:
(06/04/2022, 03:17 AM)Catullus Wrote: For an ordinal alpha greater than one, f $[Image: svg.image?\alpha]$ (x) in the Fast-growing hierarchy is greater than or equal to 2 ( $[Image: svg.image?\alpha]$ +1)-ated to the x. So should not the super function of 2 circulated to the x be less than f $[Image: svg.image?\omega]$ (x)? But on the Googology Wiki page for circulation says that "Using only nonnegative integers, there are only four cases where circulation is defined:".

Could you post a link to what a circulation is? Sorry, I'm unaware of that. Fast growing hierarchy's usually don't get much attention on this forum, it might be better to post it on a forum more geared towards that. I'm interested in what exactly this means though.

Regards, James

https://googology.fandom.com/wiki/Circulation

RE: Circulation and the Fast-Growing Hierarchy - MphLee - 06/06/2022

If I'm not mistaken, Geisler's circulation coincides with what in the Rubtsov-Romerio's terminology call omegation.
These concepts are defined via limits-convergence.

Fast growing hierarchies, instead, are part of the study of subrecursive hierarchies of function, a segment of recursion theory that deals with refining the Grzegorczyk classification of primitive recursive functions using other kind of recursion schema built around ordinal sequences of "benchmark functions" extending Ackermann-like ones.
We can say that this can be seen as an extension of the theory of goodstein maps from natural ranks to countable-ordinal ranks $\alpha<\omega_1$.

Also the FGH are constructions used in the study of definability, a part of recursion theory/computation theory: in particular in the form of ordinal notations and ordinal definability.

RE: Circulation and the Fast-Growing Hierarchy - JmsNxn - 06/06/2022

(06/06/2022, 12:42 AM)MphLee Wrote: If I'm not mistaken, Geisler's circulation coincides with what in the Rubtsov-Romerio's terminology call omegation.
These concepts are defined via limits-convergence.

Fast growing hierarchies, instead, are part of the study of subrecursive hierarchies of function, a segment of recursion theory that deals with refining the Grzegorczyk classification of primitive recursive functions using other kind of recursion schema built around ordinal sequences of "benchmark functions" extending Ackermann-like ones.
We can say that this can be seen as an extension of the theory of goodstein maps from natural ranks to countable-ordinal ranks $\alpha<\omega_1$.

Also the FGH are constructions used in the study of definability, a part of recursion theory/computation theory: in particular in the form of ordinal notations and ordinal definability.

Lol. I thought this was one of those $F_\omega$ (or whatever) functions they have in all that hubaloo. It looked like one of those. I guess it's not.

I had to look up omegation, and going off this paper https://math.eretrandre.org/tetrationforum/attachment.php?aid=222, so now I guess, it would look like this:

\[

\alpha[\omega]x = \lim_{n\to\infty}\,\alpha \uparrow^n x\\

\]

I don't know much about that, other than for $1 \le \alpha \le \eta$:

\[

\alpha[\omega]x = \alpha\,\,\text{for}\,\,\Re x \ge 1\\

\]

I think it'll probably have a nontrivial area in the complex domain where this expression converges in $\alpha$, from there it probably just equals $\alpha$.

I don't see how that would relate to Fast Growing Hierarchies. Could you clarify your question, Catullus?

RE: Circulation and the Fast-Growing Hierarchy - Catullus - 06/06/2022

(06/06/2022, 01:24 AM)JmsNxn Wrote: I don't see how that would relate to Fast Growing Hierarchies. Could you clarify your question, Catullus?

$[Image: svg.image?f_\alpha(x)%3E]$ 2[alpha+1]x. for alpha greater than 1. So $[Image: svg.image?f_\omega(x)]$ would be greater than 2[omega+1]x. the super function of 2[omega]x or 2 omegated to the x. But on the Googology Wiki page for circulation (https://googology.fandom.com/wiki/Circulation) it says "Using only nonnegative integers, there are only four cases where circulation is defined:". Should not its super function at base two grow slower than $[Image: svg.image?f_\omega(x)]$ ? $[Image: svg.image?F_\omega(x)]$ is defined using for lots of cases, using only positive integers.

RE: Circulation and the Fast-Growing Hierarchy - JmsNxn - 06/06/2022

(06/06/2022, 02:33 AM)Catullus Wrote:
(06/06/2022, 01:24 AM)JmsNxn Wrote: I don't see how that would relate to Fast Growing Hierarchies. Could you clarify your question, Catullus?
F $[Image: svg.image?\alpha]$ (x) > 2[ $[Image: svg.image?\alpha]$ +1]x. for alpha greater than 1. So f $[Image: svg.image?\omega]$ (x) would be greater than 2[ $[Image: svg.image?\omega]$ +1]x. the super function of 2[ $[Image: svg.image?\omega]$ ]x or 2 omegated to the x. But on the Googology Wiki page for circulation (https://googology.fandom.com/wiki/Circulation) "Using only nonnegative integers, there are only four cases where circulation is defined:". Should not its super function at base two grow slower than f $[Image: svg.image?\omega]$ (x)? F $[Image: svg.image?\omega]$ (x) is defined using for lots of cases using only positive integers.

OHHHHHHH!!!!

This is a very fascinating question. I will gladly answer this. It took me awhile to understand what you meant. So let's write $\uparrow f$, to represent "take the super function" of $f$. The answer to your question is actually pretty dumb, as it comes from analysis. It's only interesting in iteration theory.

Let's take $f$ and it's superfunction $F$, such that $F(s+1) = f(F(s))$. Now, let's assume $f$ is constant. $f=C$. Well then, $F=C$.

\[
\uparrow \text{Constant} = \text{Constant}\\
\]

So what happens in this omegation instance, is that we either hit $4$ when $x=2$, we hit $1$ when $x=1$, we hit $1$ when $y=0$, or we hit $\infty$. The value $\infty$ in this case can be thought of as a constant. And:

\[
\uparrow \infty = \infty\\
\]

So all of these "omegations" are hitting a fixed point value. I think what you are trying to look at is a little bit different. You want to look at:

\[
2 \uparrow^{2\uparrow n}n\\
\]

And you are asking of super functions in this manner. That's the only way I can think of which introduces fast growing hierarchies...

I hope I'm in the ball park. It'd help if you worded out more what your asking... I half get what you are asking. Can you just write more, and explain further?

Regards, James

RE: Circulation and the Fast-Growing Hierarchy - Catullus - 06/06/2022

(06/06/2022, 04:32 AM)JmsNxn Wrote: I hope I'm in the ball park. It'd help if you worded out more what your asking... I half get what you are asking. Can you just write more, and explain further?

Regards, James

2[omega+1]x would be less than $[Image: svg.image?f_\omega(x)]$ . Like 2[omega+1]3 would be less than $[Image: svg.image?f_\omega(x)]$ . And be defined.

RE: Circulation and the Fast-Growing Hierarchy - JmsNxn - 06/06/2022

(06/06/2022, 04:37 AM)Catullus Wrote:
(06/06/2022, 04:32 AM)JmsNxn Wrote: I hope I'm in the ball park. It'd help if you worded out more what your asking... I half get what you are asking. Can you just write more, and explain further?

Regards, James
2[ $[Image: svg.image?\omega]$ +1]x would be less than f $[Image: svg.image?\omega]$ (x). Like 2[ $[Image: svg.image?\omega]$ +1]3 would be less than f $[Image: svg.image?\omega]$ (3). And be defined.

I'm confused. Can you please elaborate further...?

RE: Circulation and the Fast-Growing Hierarchy - Catullus - 06/06/2022

(06/06/2022, 04:48 AM)JmsNxn Wrote:
(06/06/2022, 04:37 AM)Catullus Wrote:
(06/06/2022, 04:32 AM)JmsNxn Wrote: I hope I'm in the ball park. It'd help if you worded out more what your asking... I half get what you are asking. Can you just write more, and explain further?

Regards, James
2[ $[Image: svg.image?\omega]$ +1]x would be less than f $[Image: svg.image?\omega]$ (x). Like 2[ $[Image: svg.image?\omega]$ +1]3 would be less than f $[Image: svg.image?\omega]$ (3). And be defined.

I'm confused. Can you please elaborate further...?

$[Image: svg.image?f_\omega(3)=2%5E%7B2%5E%7B2%5E3*3%7D*2%5E3*3...21,210,694]$ . So should not 2[omega+1]3 be less than 2^402,653,184*402,653,184?