Circulation and the Fast-Growing Hierarchy

[MphLee]

Why is the operation a circulated to the b not well defined?

Because for some \(a,b\) the sequence \(\lim a\uparrow^n b\) diverges. A binary function \(f:\mathbb N\times \mathbb N\to \mathbb N\) defined as  \(f(a,b)\,:=\lim a\uparrow^n b\) DOESN'T EXISTS.
There is a partial function \(f:\,P\to \mathbb N\) where \(P\subseteq \mathbb N\times \mathbb N\) s.t. if \((a,b)\in P\) \(f(a,b)=\lim a\uparrow^n b\).

In that case: "Two circulated to the two still equals four." is correct but

Two circulated to the five may be some large infinity. Still defined.

Is something we cannot claim. Unless we define it like that. But if we do it, we must decide if we want it to be an ordinal or a cardinal. Let's assume we extend \(f:\,P\to \mathbb N\) to \({\bar f}:\mathbb N\times \mathbb N \to {\bf card}\) then we have a class function.... we still need to allow the argument to be a cardinal in order to take superfunction of that.

I fear there will be many non equivalent ways to do this. Also, the previous argument disqualifies \(f\) to be part of some fast growing hierarchy, by definition.

\(|\alpha|\leq\aleph_0\) does not hold for all countable ordinals. 

Yes it does, by definition of countable.

\(|\alpha|\leq\aleph_1\) does hold for all countable ordinals.

Nope it doesn't. By definition of uncountable and of \(\omega_1=\aleph_1\).
Conterproof. let \(\alpha=\omega_1\), then \(|\alpha|\leq\aleph_1\) is true but \(\alpha\) is uncountable.

Not all of the levels of the Fast-growing hierarchy are recursive. Not all countable ordinals have computable fundamental sequences even if they recursive ordinals.

Sure, it's what I was trying to say in my previous post.

It might be interesting to extend the hyper-operation hierarchy to transfinite ordinal ranks.

Yea, I believe that fast growing hierarchies are a tool that can be used to achieve this. If I recall correctly, I'd better check the literature, but someone important already tried to extend the lower hyperoperations to ordinals... was it by Doner and Tarski? I'll double check if needed... i have the paper somewhere.

I wonder how omega minus one would behave in some sort of extension of the Fast-growing hierarchy. I know omega minus one is not an ordinal. But it is a Hyperreal number.

This would be cool. But I'm not sure that this count as an extension of some FGH. The reason is, Hyper-reals do not extend ordinals, are kind orthogonal to them. For example, ordinal addition is a non-commutative monoid. Hyperreals under addition are an abelian group.

Defining families of hyperoperations over other number system (matrix algebras, surreals, hyperreals, tropical fields, cardinals, ordinals... and so on) sounds pretty exciting and something we should explore.