The Many and the One, Ch. 4

In the next part of their book, ‘Comparisons’, F&L discuss ‘Plurals and Set Theory’ (Chapter 4). ‘Plurals and Mereology’ (Chapter 5), and ‘Plurals and Second-order Logic’ (Chapter 6).

Here, in bald outline, is what happens in Chapter 4.  §4.1 describes a ‘simple set theory’ framed in a two-sorted first-order language, with small-x quantifiers running over a domain of individuals and big-X quantifiers running over sets of those individuals. The two sorts are linked by an axiom scheme of set comprehension, (S-Comp): ∃Xx(xX ↔ φ(x)). §4.2 notes that the mutual interpretability of this theory with a certain simple plural logic. (We can’t simply replace big-X set variables by double-x plural variables, at least given the usual assumption that there is an empty set in the range of big-X variables but not an empty plurality in the range of double-x plural variables. But working around that wrinkle involves only minor tinkering.) §4.3 then asks whether this mutual interpretability means we should eliminate plurals in favour of sets or alternatively eliminate sets in favour of plurals. §4.4 suggests that we need plurals in elucidating the very notion of a set (so don’t eliminate plurals): the root idea is that “For every plurality of objects xx from [a given domain], we postulate their set {xx},” where postulation seems to be tantamount to defining into existence. We are promised more about definitions of this kind in Chapter 12.

§4.5 then notes that mathematical uses of sets crucially involve not just sets of individuals (numbers, perhaps) but sets of sets, sets of sets of sets. etc.; and, for a start, it is very unclear that these can be eliminated in favour of pluralities of pluralities. §4.6 then says more about the iterative conception of set, and §4.7 gives the axioms of ZFC. §4.8 jumps on to wonder whether we can use plurals in explicating the notion of proper classes. The chapter ends with §4.9 which raises a problem:

We have described two very attractive applications of plural logic: as a way of giving an account of sets, and as a way of obtaining proper classes “for free”. Regrettably, it looks like the two applications are incompatible. The first application suggests that any plurality forms a set. Consider any objects xx. Presumably, these are what Gödel calls “well-defined objects”. If so, it is permissible to apply the “set of” operation to xx, which yields the corresponding set {xx}. The second application, however, requires that there be pluralities corresponding to proper classes, which by definition are collections too big to form sets.

F&L again promise to return to deal with this apparent tension in their Chapter 12.

Does the chapter work? Well, although I said in my first post on the book that I wouldn’t fuss too much about this sort of thing, it is pretty difficult to know quite at whom this chapter is aimed. For example, §4.6 very briskly outlines the iterative conception of set, helping itself along the way to the idea that we take unions at levels indexed by limit ordinals (where ordinals are unexplained). But I wonder who is supposed to (a) already be familiar with the notion of a limit ordinal in §4.6, but (b) still need to have the axioms of ZFC given again in §4.7? And won’t the reader who actually needs §4.7 then need more explanation of the role of proper classes in set theory (and the difference between their appearance as virtual classes in e.g. Kunen, versus a more substantive appearance in NBG)?

And to go back to the beginning of the chapter, I would guess that someone with enough logical education to know about limit ordinals would also know enough to want to ask more about the principle S-Comp: does the comprehension principle apply to predicates φ(x) which themselves involve bound set variables? or involve free set variables as parameters? or neither? We are not told, and there is no hint that the issue might matter. There is also no hint at all that the kind of “simple set theory” with two sorts of quantifier might actually be of real interest, e.g. in reverse mathematics when considering subsystems of second-order arithmetic. This lack of development is typical and disappointing.

As it happens, I am in sympathy with F&L’s overall line that (i) plural logic is repectable and can earn its keep in certain important contexts, and (ii) set theory is just fine in its place too! But I can’t see that this arm-waving chapter really advances the case for either limb (and I could nag away more at some of the details). In so far as there are hints of novel argumentative moves, the work of elaborating them is left for much later. So I did find the level of discussion in this chapter frustratingly rather superficial: hopefully, F&L do better when they return to cash out those promissory notes.

To be continued.

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