On now to the second half of the second chapter, where we are still considering the iterative conception in an initial way.
So, quoting Luca,
According to the iterative conception, then, sets can be arranged in a cumulative hierarchy divided into levels. This conception sanctions (at least) most of the axioms of standard set theory and provides a convincing explanation of the paradoxes; but is it correct?
What reasons, then, can be offered in support of endorsing the iterative conception?
Luca first discusses the idea that we should take literally the metaphor of construction that comes to us so readily in describing the iterative conception. So, the idea is, sets really are formed in a stage-by-stage process, where at each stage we can only collect together in various ways what is already available.
But how do we make better-than-metaphorical sense of this idea of forming sets in a process when we are supposed to be dealing (aren’t we?) with abstract items which (i) exist independently of our activities (aren’t really formed) and (ii) in a timeless way (so there’s no real process of level-building). Arguably, the constructionist metaphor at best gives colour but no real underpinning to the iterative conception.
Suppose, however, we do try to push the metaphor harder. Then, Luca argues, [my numbering]
(i) it seems part of the constructivist doctrine that, at any point in the construction process, we can only construct sets specifiable by reference to sets already constructed. (ii) This, however, seems to sanction only a predicative version of Z’s Separation Schema …
which cuts down the strength of our set theory. Now, (i) gives us one way of elaborating what the ‘the constructionist doctrine’ might be supposed to be. Though we could, I suppose, pause to ask whether is it compulsory to construe ‘construct sets from sets that are already constructed’ as implying ‘construct sets specifiable by reference to sets already constructed’. Be that as it may, it would have been good if Luca had then paused longer over the implications of (ii), saying more about predicative set theories. Just how weak are they? If we can live with weak predicative set theories for ordinary mathematical purposes (as Feferman claimed, for example), then why not treat them as what is, on second thoughts, warranted by a rather strictly interpreted iterative conception? Some readers might have wanted rather more here.
However, with the iterative conception so understood, we’ll have to back off from our original thought that the iterative conception sanctions (most of) standard set theory. And Luca takes this in itself to be a reason to resist the constructivist gloss on the iterative conception.
Moving on — we’ve got to §2.4 of the book — Luca next considers the idea that we can underwrite the iterative conception, not by saying that the sets are literally ‘formed’ stage by stage, but by invoking a [now timeless] relation of metaphysical dependence between a set and its members: the hierarchy reflects this structure of metaphysical dependence.
Not surprisingly — or at least, not surprisingly to someone as sceptical about such metaphysical notions as I am — Luca has little trouble in showing that various attempts to elucidate this supposed relation of metaphysical dependence in terms of other metaphysical notions (like that of essential property) either go round in very tight circles, or pretend to explain the obscure in terms of the even more obscure. Moreover it is quite unclear, as Luca also argues, that even if we could make good a suitable notion of metaphysical dependence, that this would underpin an iterative hierarchy of the right structure (can’t there be, for a start, circles of metaphysical dependencies?). The critical discussion in §2.4 seems pretty conclusive to me.
So where does that leave us? We can’t, it seems, underwrite the iterative conception (or at least an iterative conception that will sanction something like standard set theory) by trying to cash-out a construction metaphor or a metaphor about dependence. But then recall this well-known remark from George Boolos about the iterative conception, aptly quoted by Luca:
[F]or the purpose of explaining the conception, the metaphor is thoroughly unnecessary, for we can say instead: there are the null set and the set containing just the null set, sets of all those, sets of all those, sets of all Those, … There are also sets of all THOSE. Let us now refer to these sets as ‘those’. Then there are sets of those, sets of those, … Notice that the dots ‘…’ of ellipsis, like ‘etc.,’ are a demonstrative; both mean: and so forth, i.e. in this manner forth.
Luca picks up on Boolos’s thought, and argues that we should indeed be content with what he calls a minimalist account of the iterative conception (we are supposed to hear echoes here of talk about a minimalist account of truth — I’m not entirely convinced that’s helpful, given that minimalism about truth is deflationist in spirit while Luca’s iterative conception remains very robust; but let that pass). He finds such a conception already in Gödel, quoting a remark where he talks of a concept of set
according to which a set is anything obtainable from the integers (or some other well-defined objects) by iterated application of the operation (“set of”).
And that, the suggestion goes, is where the iterative conception bottoms out, just in the idea of iterating applications of ‘set of’ (where the result of an application is distinct from any of the things it is applied to).
Note, in passing, that if what crucially matters is the set of operation, and (as Boolos’s words indicate) this operation takes zero, one, or more things (plural), and yields a set of them, then arguably the natural logical home for set theory would seem not to be standard first-order logic which has no place for plurals (no formal equivalent of ‘those’!); rather it seems we will want a plural logic which can treat operations mapping many to one. We’ll have to see if this thought is taken up later.
Anyway, Luca proposes that we take the iterative conception ‘neat’ (so to speak), without the supposed support of further thoughts about constructions or dependencies. But without those illusory further supports, why should we think it is a good conception? Well, this is what the rest of the book is going to be about … showing on the hand that other conceptions won’t give us what we want, and on the other hand that the iterative conception, minimally construed, can resist various critical attacks. So we get (in Luca’s words) an ‘inference to the best conception’. We’ll have to see how this promised inference pans out!
To be continued: the next chapter is on Challenges to the Iterative Conception.