What led Hausdorff to consider the possibility of "unerreichbar" or "weakly inaccessible" cardinals?

Question

I would like to know how it was that Felix Hausdorff came to consider the possibility of, if not indeed necessity of, what he was the first to call "unerreichbare" or "unreachable" cardinals, or what we call, somewhat less clearly, "[weakly] inaccessible cardinals" today. To me such a move clearly constitutes a huge leap of imagination, much like the earlier one Cantor made when he proved that not all infinite sets have the same size. [even if that should have been clear already from the elementary proof that there can be no bijection between ANY set and its powerset] I suspect there was something in Hausdorff's very rich research project, mostly on the ordinals at the time, that "compelled" him to hypothesize such, to his day, very "large" cardinals; indeed cardinals much larger than, and "unreachable" from, the entire series of Beth cardinals, [at least from Beth1 to Bethω as were already known at the time]? ... I also haven't yet found much written about what the reception of Hausdorff's unreachable cardinals was by his contemporaries. I assume many of the latter must have thought the notion completely preposterous, but even that I haven't been able to confirm. Any direction anyone can provide to me about such things will be greatly appreciated!

Answer 1

As bof said, I think the notion is actually natural enough that it would be weird to not ask about it. But in fact there is a good reason for Hausdorff to explicitly consider it.

The big question in set theory at the time was of course the continuum hypothesis. In $1904$, Konig had proved an interesting technical result about the continuum, namely $$cf(2^\omega)>\omega$$

(actually he proved something more general). Indeed this was the only nontrivial result about the continuum's cardinality known when Hausdorff wrote his book in $1908$, and as Cohen would observe later the only nontrivial result provable in $\mathsf{ZFC}$ at all. Moreover, Hausdorff had a role in this result: it was extracted from a failed argument against $\mathsf{CH}$ put forth by Konig, whose error was detected by Hausdorff (and independently Konig himself). See here for more on this.

So by the time Hausdorff is writing in $1908$, there's already a known connection between the biggest question around and the notion of cofinality, and Hausdorff himself has been involved in forming this connection. Understanding the role of cofinality in cardinal arithmetic is therefore a very natural topic for him to think about, and among the first questions which arise is: "Are the uncountable regular cardinals exactly the uncountable successor cardinals?" So right there the notion of weak inaccessibility emerges naturally.

(That said I don't know of a source which definitely says that this is why Hausdorff introduced them. But it was definitely "in the air" at the time.)

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It's also worth noting that a certain amount of the idea "weakly inaccessible cardinals are really big" took a while to become clear. Specifically, slogans like

weakly inaccessible cardinals are so big we can't prove they exist

are rather common, even if they're fairly misleading (e.g. since we can have $2^\omega$ be weakly inaccessible if weakly inaccessibles are consistent in the first place). Now remember that the idea behind such slogan comes from combining two theorems of Godel, and one communal agreement:

1. $\mathsf{ZFC}$ captures "what we can prove about set theory."

2. $\mathsf{ZFC}$ cannot prove that $\mathsf{ZFC}$ has a model (let alone a transitive model!) unless $\mathsf{ZFC}$ is inconsistent.

3. If $\alpha$ is weakly inaccessible, then $L_\alpha\models\mathsf{ZFC}$.

But $\mathsf{ZFC}$ wouldn't even be introduced (let alone essentially universally accepted) in its modern formulation until the late $20$s, while Godel wouldn't prove $(2)$ and $(3)$ until the early $30$s and $50$s respectively.

• OK fine, we can easily sidestep $(2)$ above: it would be enough to show instead that $L_\alpha\models$ "There are no weakly inaccessible cardinals" if $\alpha$ is the least weakly inaccessible cardinal, which follows immediately from the appropriate absoluteness of weak inaccessibility. But as far as I'm aware of, even though it's easier that sort of argument post-dated the second incompleteness theorem, so the point about historicity still stands. EDIT: it turns out that my history is a bit off; absoluteness proofs that $\mathsf{ZFC}$ doesn't prove the existence of a strongly inaccessible cardinal were given in the $20$s. Still, this significantly postdates Hausdorff's work.

And finally, observe that point $(3)$ is actually surprisingly subtle. For strong inaccessibility things are very simple: it's almost trivial to show $V_\alpha\models\mathsf{ZFC}$ whenever $\alpha$ is strongly inaccessible. So as soon as we think about absoluteness, or learn the second incompleteness theorem, it's obvious that strongly inaccessible cardinals reach beyond $\mathsf{ZFC}$. However, this argument breaks down for weak inaccessibility: it need not be the case that $V_\alpha\models\mathsf{ZFC}$ if $\alpha$ is weakly inaccessible, and in fact - as noted above - it's possible that $2^\omega$ itself is weakly inaccessible. Instead, some genuine work is needed here: I don't offhand see a way to show that weakly inaccessible cardinals imply the consistency of $\mathsf{ZFC}$ without developing the basic theory of $L$, which is highly nontrivial.

So our modern understanding of the logical strength of weak inaccessibility wasn't available to Hausdorff in $1908$. Of course it's clear from the definition that weakly inaccessible cardinals are pretty big, but there isn't that much evidence at the time that they're particularly special. So I suspect that their reception was much more muted than a reader brought up on the modern story of large cardinals might expect.