Original purpose of Fundamental Sequences of limit ordinals

Question

I am curious of why fundamental sequences of limit ordinals were invented? Was it only to be able to define a function (e.g. fast-growing) hierarchy?

For instance:

• Zero ordinal 0: $f_{0}(n) = n + 1$
• Successor ordinal $\alpha + 1$: $f_{\alpha+1}() = f^n_{\alpha}(n)$
• Limit ordinal $\lambda$: $f_{\lambda}(n) = f_{\lambda[n]}(n)$, where $\lambda$ is a limit ordinal and $\lambda[n]$ is the nth ordinal of the fundamental sequence of $\lambda$. Is it because of this case that fundamental sequences were invented? To make it possible to reduce limit ordinals to successor ordinals to ensure that the recursion terminates? And also because when the argument $n$ increases, then the nth ordinal of $\lambda$ increases as well, yielding a faster growing function?

Or was it for any other reason? Which reason in that case?

Do you know any references regarding the fundamentals and the history of function hierarchies and fundamental sequences of limit ordinals?

Jonas

Answer 1

As far as I can tell, the first appearance of fundamental sequences was (sort of) in Hardy's $1904$ paper "On a theorem concerning the infinite cardinal numbers." Here Hardy essentially introduced the notion of a fast-growing hierarchy, in the process of constructing an embedding of $\omega_1$ into the reals (viewed as functions from $\mathbb{N}$ to $\mathbb{N}$).

If I recall correctly however (I don't have a copy of the paper on hand), Hardy didn't actually use fundamental sequences!$^*$ That is, he didn't seem to take into account that each limit (diagonalization) step of his construction required a choice of fundamental sequence for that limit ordinal. I think this was only observed later.

So tentatively, I believe:

Fundamental sequences were first implicitly used in Hardy's proof that $\omega_1$ injects into $\mathbb{R}$, although he did not actually use them explicitly (and may not have recognized their necessity). This argument wound up constructing a fast-growing hierarchy, although that wasn't the actual purpose at the time.

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$^*$This is indirectly supported by this expository paper; however, since it doesn't directly quote the article in question, I can't fully vouch for it until I get a chance to take a look at Hardy's actual paper.