Complex Hardy Hierarchy

[Catullus]

How could the Hardy hierarchy be defined for complex numbers? Like, what are \(H_{\omega^\omega}(i)\) and\(H_{\varepsilon_0}(i)\) with respect to the Wainer Hierarchy system of fundamental sequences and \(\varepsilon_0[n]={}^n\omega\)?

[MphLee]

Do you have any idea about how to go for it?
How would you address the fact that the fundamental sequence concept and the transfinite constructions is based on the well order structure of ordinals while complex numbers are a number system that is inherently un-orderable in a compatible algebraic way?

[Catullus]

Do you have any idea about how to go for it?
How would you address the fact that the fundamental sequence concept and the transfinite constructions is based on the well order structure of ordinals while complex numbers are a number system that is inherently un-orderable in a compatible algebraic way?

 My idea is that it may be possible to do complex iteration on the function that maps \(f_\alpha\) to \(f_{\alpha+1}\), using a similar uniqueness criteria to this uniqueness criteria for tetration. And then, once we have \(f_\omega\) defined for complex numbers, we could again do complex iteration on the function that maps \(f_\alpha\) to \(f_{\alpha+1}\), and define \(f_{\omega*2}\) for complex numbers. And then, doing iteration on the function that maps \(f_{\omega*\alpha}\) to \(f_{\omega*(\alpha+1)}\), \(f_{\omega^2}\) could be defined for complex numbers.
You could keep iterating and iterating iteration, and so on, until you eventually do complex iteration on the function that maps \(f_{\omega\uparrow\uparrow\alpha}\) to \(f_{\omega\uparrow\uparrow(\alpha+1)}\), and define \(H_{\varepsilon_0}\) for complex numbers!!!!
Of course, this sound ludicrously difficult.

Just because complex numbers are inherently un-orderable in a compatible algebraic way, does not mean they can not be put into ordinal hierarchies. For example, in the slow-growing hierarchy, \(g_{\varepsilon_0}(z)=z\uparrow\uparrow\uparrow2\), which is definable for complex numbers.

[MphLee]

Maybe it is possible to complex iterate \(f_\alpha\mapsto f_{\alpha+1}\)... but how can we? Many times on this forum it was repeated that this is the key problem.

Anyways Catullus, I encourage you to make a lil extra step , try to propose your formal definition instead of just vague ideas. Everyone here will benefit of it.