We have now reached Chapter 3, which can be conveniently divided into three parts. The first part (§§3.1–3.2) discusses two initial, and very general, challenges to the iterative conception, challenges which (it seems to me) can be fairly readily met. These are the topic of this post. The second part of the chapter (§§3.3–3.5) discusses another very general challenge, to my mind a rather more interesting one: I’ll consider that in my next post. The final part (§3.6) discusses a more specific challenge (meaning one that arises from focused technical questions, about the status of replacement, rather than from sweeping conceptual considerations). I’ll need to revise my homework on replacement! — but hope to have something sensible to say in a third post. So to begin …
The first challenge to the iterative conception we’ll discuss is what Luca calls the missing explanation objection. In brief,
[I]f we take all sets to be those in the hierarchy, we cannot explain the appeal of the naïve conception of set, as embodied in the Axioms of Comprehension and Extensionality.
This supposed objection has been pushed by Graham Priest, but it has always struck me as pretty feeble. Assume we have distinguished the concept of set (a unique object over and above its members) from the idea of a class-as-many. Now we have this intended concept of set in play, there is room to further distinguish the following two claims: (i) [Naive property comprehension] for any property, there is a set of all and only the objects with that property, and (ii) for any determinate plurality of objects, there is a set of all and only them. Now, (i) gives us e.g. a set of all sets, while (ii) doesn’t — because no determinate plurality of sets is all the sets (since given that plurality of sets there is, by (ii), another set, namely the set of them). The defender of the iterative conception, who will reject (i) but can accept a version of (ii), can then reasonably say that once we’ve distinguished sets from classes-as-many, the remaining appeal of (i), such as it is, comes from confusing it with (ii). And once the distinction is made and properly grasped, the appeal should vanish. [Of course, this isn’t to say that the iterative conception is definitely right, or that the idea of a universal set is definitely wrong: the challenge though was that the iterative theorist had nothing to say about the appeal of the naive conception — and that seems wrong.]
This, at any rate, is the sort of story I would have given. Luca says rather more over six and a half pages. In the first past of his discussion, he presses the distinction between (i) and the claim (i*) for any property of individuals, there is a set of all and only the individuals with that property [where the individuals are the non-sets]; and he suggests that part of the appeal of (i) comes from confusing it with, or recklessly generalizing from, the harmless (i*). Perhaps there is something in that, though I’m not very sure. The second part of Luca’s discussion then gives a more careful treatment related to — though not, I think, quite the same as — the response that I sketched.
The second challenge to the iterative conception, again pressed by Priest but also encountered elsewhere is what Luca calls the circularity objection. This arises from the suggestion that iterative conception of the set-theoretic is “parasitic on a prior notion of an ordinal” and, if we are not going to go round in circles, that’s will need to be derived from a different notion of set (so the iterative conception can’t be fundamental).
But this too has always struck me as feeble (roughly: it depends on forgetting that the von Neumann ordinals are a handy implementation, not the one-and-only possible story about ordinals-as-indexers-for-tranfinite-processes). After all, we can get a long way into the theory of at least countable ordinals without talking about sets at all — we just need numbers (as individuals) and order-relations on them. If you insist on treating relations as sets of pairs which are themselves sets of sets, you still only need a few levels of sets. So: start with the numbers and a few levels. Develop a theory of countable ordinals. Use them to index more levels (lots of levels!) to get a very rich universe. In this universe we can define many more ordinals. OK, so we can now lever ourselves us by indexing more levels with these new ordinals. And so on upwards … There’s no circularity. When adding stages of the hierarchy, we already can define the ordinals we need to index the additional stages. This sort of idea was already being explained by Gödel in 1933.
Ok, that’s a bit arm-waving, but they basic idea is probably familiar. Turning to Luca’s discussion, he first gives a considerably more careful and more developed version of this Gödelian pre-emptive response to the challenge.
But he then adds a very important additional point which is worth highlighting here:
The axiomatization given by Scott (1974), of which [Scott-Potter] SP is a descendant, shows that the worry that the notion of a well-ordering is needed to grasp the iterative conception is really just an idle concern. In particular, what Scott provided is an axiomatization of set theory which, albeit sanctioned by the iterative conception, does not assume a previous conception of the hierarchy as constituted by levels ordered by the ordinals. Rather, starting from certain elementary facts about levels, which … he called ‘partial universes’, he established facts about sets and levels. Notably, what is assumed about the levels does not include that the levels are well-ordered. More specifically, he showed that, assuming the Axioms of Restriction and Accumulation, we can prove, together with the Axioms of Separation and Extensionality, that all axioms of Z except for Infinity hold, that every set is well-founded and, crucially, that the levels are well-ordered by membership. … The upshot is that the fact that the levels of the hierarchy are well-ordered is not required to grasp the iterative conception, but is a consequence of it. I conclude that we do not need a prior and different notion of set to make sense of the notion of the cumulative hierarchy, and the circularity objection fails.
That seems conclusive.
Reading §§3.1–3.2, I found myself wondering how much the intuitive appeal of the naive conception, and of the unhappiness with standard set theory, is because of what’s essentially a historical accident: that the naive conception arrived first.
First at least in the usual rational reconstruction, for that goes something like this: Our understanding of sets began with the appealing and intuitive idea — only later characterised as “naive” — that for any property / predicate, you could form a set of all things that had that property / satisfied that predicate. Paradise beckoned. Then Russell found the worm in the apple. Disaster! Set theory fell in ruins. Next followed a rather ad hoc attempt to cobble together enough of the ruins to form something that could handle as much set theory as was needed to do mathematics and that looked like it would at least avoid the paradoxes we already knew about.
Seen that way, set theory was pretty much bound to be seen as questionable and artificial, in need of something that would give coherence to its axioms and make them seem reasonable.
But what if things had happened in a different order? What if something like the iterative conception came along first. As Godel pointed out
If someone then suggested there should be unlimited, any-property comprehension, Russell’s paradox would have shown the idea was inconsistent, it would have been abandoned, and we’d have continued happily using the set theory we already had.
I think I rather like that!
The response that is sketched in the blog to the ‘circularity objection’ may be fine as far as it goes, but I don’t think that it gets to the bottom of the matter. At best, it shows that we can describe the cumulative hierarchy in a way which, while appealing to sets and ordinals, assumes rather less about them than is declared in Z; so it is not fully circular. But something more fundamental seems to be at stake, and one should not be afraid to grab the bull by the horns. We already know that the only way to give a straightforward account of the semantics for truth-functional connectives is by using such connectives, explicitly or implicitly, in our metalanguage, and that we cannot present the semantics for first-order quantifiers, in either the x-variant or the substitutional version, without using quantifiers themselves. We can’t even define the languages for those two logics without using conditionals, conjunction, universal quantification and some sets. Yet those accounts still provide us with satisfying intuitions about what classical logic is all about. Likewise, the picture offered by the cumulative hierarchy is illuminating and helpful even if it has residual elements of circularity. After all, the goal is not to prove, out of nothing, that ZFC or even just Z is correct. That is an unrealizable pipe-dream. The goal is to make good intuitive sense of what is going on, to get our heads around the basic ideas. If the account succeeds in doing that, circularity is not a problem. Personally, I think that the picture of sets that isprovided by talk of a cumulative hierarchy achieves its goal only partially, since it does not motivate the axioms of replacement or choice; but it is the best single motivating story that we have at present.
Sure — as with any fundamental web of concepts — our story has to bottom out somewhere, and our elucidatory chat will then have to go round in what, we hope, are helpfully illuminating circles. But I’m still inclined to say that it would be Bad News — wouldn’t it? — if in explaining the iterative conception we had to call on an understanding of the ordinals which already presupposes we have a fully-fledged set theory up and running (we’d be explaining what was supposed to be the less theoretical motivation by appeal to a more theoretical bunch of ideas which need motivating). So it is worth showing as Luca does that (and Gödel before him!) that we need not accept the supposed Bad News.