Rehabilitating Zermelo's description of the universe of sets?

Question

I was reading various essays about Cantor's doctrine of absolute infinity, and it came up again that Zermelo's doctrine, by contrast, was of V as an "unfinished totality." Initially, this seemed inconsistent of Zermelo: in his account of the Axiom of Choice, for instance, he objects to Cantor's account of the same on the grounds that Cantor seems to draw on temporal intuition too much.

So one interpretation of Zermelo that I found was, "V as a potentially infinite sequence of actual infinities." This won't do, though: for there are arbitrarily many actually infinite subsequences of the overarching sequence (e.g. all the alephs from ℵ₀ to ℵ_ω). I think, in other words, that if there were not absolutely all the alephs given, but only an unfinished initial segment as such, then the cofinality of V would be less than V. Granted, if you work in ZFC sans replacement, you have ℵ_ω as a placeholder for V, which means that you sort of have cf(V) = ℵ₀. Something similar can be "done with" worldly cardinals, too.

On the other hand, if inaccessible cardinals are motivated by the reflection principle, then we have that V must be equal to its cofinality (i.e. would be regular, if it had a cardinality).

Now, I'm not saying that Zermelo self-consciously meant his talk of an "unfinished totality" to be understood in the terms with which I will spell out a possible gloss on said talk. I make no such commitments regarding my theory of literary interpretation (if you will). But so anyway, another description of Zermelian V that showed up in my reading was "incomplete." On the face of it, there's not much distance between "unfinished" and "incomplete," after all. However, if we shift our attention to the question of set theory's logical background, we find that there is logical space to construct paracomplete set theories.

So, by way of comparison, in paraconsistent set theory, we have some propositions that are true and false, or perhaps you would say pairs of conflicting but true propositions. I haven't gone over Weber's work exhaustively, but I did pick up from him that in paraconsistent set theory, we can say that ORD is indeed an ordinal, that its cardinality is ℵ_ORD, that ORD = ORD + 1, and that the cardinality of the universe is both equal to and less than its powerset.

Can Zermelian V, on the other hand, be interpreted as a paracomplete universal cardinality? So instead of saying that absolute infinities are "inconsistent multiplicities" (a phase of Cantor's viewpoint), we say that proper classes are neither equal to, nor lesser or greater than, each other: they are absolutely incommensurable. (Then von Neumann's description of proper classes is false: they are not equally large!)

A technical objection/complication: if some form of paracompleteness is a property of V, and if the reflection principle says that any specific property of V is reflected by a "set-sized" object, does this mean there are paracomplete sets? There goes the well-ordering of the universe, to some extent! However, it is not clear to me that the reflection principle covers all "properties"/predicates of V whatsoever. For example, V has the "property" of being identical to V. Now while it is true that everything is identical to itself, and so everything reflects V as such, it is of course not true at all that everything is identical to V, wherefore there is some "gap" between the sets and V such that the reflection principle only reflects the more abstract character of V, here (general self-identity), not the "concrete" self-identity of the entire universe. Likewise, maybe the logical "property" of paracompleteness would not be reflected into the sets?

What range of "theories" of absolute infinity/V/proper classes can be designed by variation over the logical background as such? For we have paraconsistent set theory say that V transcends non-contradiction, paracomplete set theory that V transcends bivalence/exclusive disjunction... If we have a logic without unrestricted modus ponens, does that give us another image of V's transcendence? Or suppose that conjunction varies, here: it is no longer true that if we have A (as a truth of V) on the one hand and B (a separate truth) on the other that we can infer that we have A-and-B? How might these variations allow for novel solutions to questions like, "Is the powerset of the universe larger than the universe?" And, of course, is it fair to interpret Zermelo in this context?

Answer 1

Conifold, thanks for the Jané reference. As I read it, my impression is that Jané has only the shakiest understanding of Cantor's work on infinities, because his discussions of the subject can be interpreted a million different ways.

But in the the end, I think I understand what Jané is saying Cantor's "absolute infinite" means. Cantor defines an ordered (technically: "well-ordered") sequence of infinities. Each one is denoted by the symbol aleph (א), with a subscript that denotes the place of this infinity in the sequence.

But this is no ordinary sequence: For any infinity in the sequence there are infinitely many larger infinities. And the number of those larger infinities is larger than any infinity in the sequence.

I believe Jané is saying Cantor's "absolute infinite" is the infinity of the collection of all the infinities in the aleph sequence. This collection is so large that mathematicians know it cannot be called a "set", even though every infinity in the aleph sequence corresponds to a set of that exact infinite size.

I'm not certain that this is what Cantor meant by "absolute infinite", however, because Jané does not describe it in the straightforward language that I have tried to use in this post.