Sunday, May 06, 2007

What are sets for?

Yiannis Moschovakis on p. 1 of his very useful Notes on Set Theory writes that one "basic property of sets" is that

Every set A has elements or members.
And then, on p. 2, he writes
Somewhat peculiar is the empty set ∅ which has no members.

But of course he can't have it both ways. Either every set has elements (and there is no empty set) or there is an empty set (and so not every set has elements).

I offer this as another example to my esteemed colleagues Alex Oliver and Timothy Smiley who have a lot of fun with this sort of thing at the beginning of their recent paper "What are sets and what are they for?" (in John Hawthorne, ed., Metaphysics), and who give lots of other examples of set theorists' arm-waving introductory chat about sets being similarly hopeless. But what are we do about that? Alex and Timothy take the stern line that we should take such set theorists at their introductory word, and if that word is confused, then so much the worse for them and for the very idea of the empty set (and for the idea of singletons too, if sets are defined to be things with members, plural). Pending a secure argument for the overwhelming utility of postulating them, we should do without empty sets and singletons and indeed without the whole universe of pure sets.

But it doesn't seem good strategy to me to take set theorists at their first word -- any more than it would be a good strategy to take quantum theorists at their first word (if that introductory word involves a metaphysical tangle about particles-cum-waves). However, I'm with Alex and Timothy that the question "What are sets for?" surely is just the right question to ask. Moschovakis indicates one sort of answer (indeed, they mention it): the universe of sets provides a unified general framework in which we can give "faithful representations" of systems of mathematical objects by structured sets. Now this is, of course, the sort of thing that category theorists aim to talk about: in their terms, there will be structure-preserving functors from other (small) categories to the category of (pure) sets -- roughly because the category of sets has such a plenitude of objects and morphisms to play with. What we need to do here is think through what this kind of use for sets -- a use that can be illuminated in category-theoretic terms -- really comes to. My hunch is that we'll get to a rather different place than Alex and Timothy.

I've been a bit slow to get thinking about set theoretic matters since being back in Cambridge (there's the decidedly daunting prospect of having Michael Potter and Thomas Forster locally breathing down my neck ...). But Alex and Timothy's provocations are hard to resist. So something else to add to the list of things to think about.

1 comment:

Enigman said...

A question I wish I knew the answer to (as I don't find the standard set-theoretic infinities to be plausibly realistic), so I shall look up the paper you mention. My initial reaction is faithful representations of systems of mathematical objects? Sometimes it seems to me that mainstream mathematicians define mathematical objects as sets. And even when they do not, insofar as they are mainstream I suspect that they would deduce the more obscure properties of those (obscure) non-set objects from the (nice) properties of the sets (which seems to amount to defining them to be sets in practice). Anyway, I look forward to reading your further thoughts. Incidentally, this is a recent look at sets by a mathematician, who says: "a big part of why set theory makes such a good basis for mathematics: it's one of the simplest things that we can use to create a semantically meaningful complete logic."