Circulation and the Fast-Growing Hierarchy

[Catullus]

For an ordinal alpha greater than one, 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 ? But on the Googology Wiki page for circulation says that "Using only nonnegative integers, there are only four cases where circulation is defined:".

[JmsNxn]

For an ordinal alpha greater than one, f(x) in the Fast-growing hierarchy is greater than or equal to 2 (+1)-ated to the x. So should not the super function of 2 circulated to the x be less than f(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

[Catullus]

For an ordinal alpha greater than one, f(x) in the Fast-growing hierarchy is greater than or equal to 2 (+1)-ated to the x. So should not the super function of 2 circulated to the x be less than f(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

[MphLee]

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.

[JmsNxn]

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/tetrationfor...hp?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?

[Catullus]

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

2[alpha+1]x. for alpha greater than 1. So 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 ? is defined using for lots of cases, using only positive integers.

[JmsNxn]

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

 F(x) > 2[+1]x. for alpha greater than 1. So f(x) would be greater than 2[+1]x. the super function of 2[]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(x)? F(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

[Catullus]

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 . Like 2[omega+1]3 would be less than . And be defined.

[JmsNxn]

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[+1]x would be less than f(x). Like 2[+1]3 would be less than f(3). And be defined.

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

[Catullus]

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[+1]x would be less than f(x). Like 2[+1]3 would be less than f(3). And be defined.

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

. So should not 2[omega+1]3 be less than 2^402,653,184*402,653,184?