06/06/2022, 01:24 AM
(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/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?
