My question is about large countable ordinal numbers and their fundamental sequence.
It looks as our knowledge of large countable ordinal number theory grows,
brand-new fundamental sequence for brand-new limit ordinal is defined, created.
Can we define the general notion of fundamental sequence for limit ordinal in advance?
I mean, without development of brand-new ordinal.
Just as Cantor defined ordinal numbers before he even knew what ordinal numbers there were.
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Can you construct a function $F$ such that
domain of $F$ = set of limit ordinals less then O
(O is a certain limit ordinal)
$F(x) =$ a strictly increasing sequence of ordinal numbers converging to $x$
F should be computable, intuitive, natural and not artificial. And for O, the larger, the better.
For simplicity, let us only consider countable ordinals. I do not think only considering this case answers your question because most googologists are mainly interested in constructing countable large ordinals. It might be a longer comment than an answer, but I hope it is helpful.
If you do not care about a fundamental sequence being canonical, then there is a way to find it: For each limit countable ordinal $\alpha$, choose a cofinal subset $C_\alpha\subseteq \alpha$. Then understand $C_\alpha$ as a fundamental sequence of $\alpha$. It looks a bit weird, but choosing such $C_\alpha$ is the starting point of Todorcevic's theory of minimal walk. Such $\langle C_\alpha\mid \alpha<\omega_1\text{ is a limit ordinal}\rangle$ is called a ladder system.
However, $C_\alpha$ we chose need not to be canonical (or natural), and I believe the axiom of choice is necessary to choose a cofinal $C_\alpha\subseteq\alpha$ for each countable limit $\alpha$ simultaneously. You may want to have canonical fundamental sequences, but the meaning of being canonical is unclear. However, we may agree that concurrent well-defined ordinal notations provide canonical fundamental sequences.
Then let us see why people defined ordinal systems. Some googologists define them just for fun, to reach into larger countable ordinals. However, proof theorists defined them for ordinal analysis. Ordinal analysis is, in short, a field reducing the consistency of formal systems (like subsystems of full second-order arithmetic or $\mathsf{ZFC}$) into the well-orderedness of a given ordinal. It is a slightly misleading explanation, and it is not the only role of ordinal analysis, but this explanation shows why proof theorists defined stronger ordinal notations: Stronger ordinal notation allows us to describe the complexity of proofs over a more complicated formal system.
However, all of formal systems we practically use is recursive (that is, there is a computer program enumerating their axioms), so the corresponding ordinal notations are also recursive. All of ordinal notation you have seen is very likely recursive, which means, they have a limit to express ordinals. The limit is the least non-recursive ordinal $\omega_1^\mathsf{CK}$, also called the Church-Kleene ordinal. There is no way to find a recursive cofinal sequence of $\omega_1^\mathsf{CK}$ since every recursive sequence of recursive ordinals has a limit less than $\omega_1^\mathsf{CK}$.
The answer might be different if we consider a different notion of being canonical (like, $E$-recursion), but I do not think the answer is positive even in those cases.