Wednesday, June 27, 2007

The Empty Set?

Standard mathematics is based upon an empty set (a set with no members), which is identified as the whole number 0. But while there is only one whole number 0 (the number of things that we have when we have nothing), there are many set theories. There is only one standard theory, but standards change (while 0 does not) and anyway, even were one particular theory especially elegant, coherent and useful, that would not make it true (cf. ideal fluids and Newtonian astrophysics). So the question arises, what could sets be if not formal? Well, what is a chess set? It is 32 chess pieces (and possibly also a checkerboard) of the right kind. If one of them is missing, then that is not a chess set, although it is still a collection of chess pieces—what logicians mean by ‘set’ is something like a collection.
......So, what are collections? Well, what is a stamp collection? It is a collection of stamps—all of the stamps collected by, owned by some individual. It does not include the stamp albums, just the stamps; and if it is sold then it remains the same collection of stamps, belonging to someone else. And if the collector had to sell most of his stamps, and had only one left, then his collection would consist of just that stamp—his collection would be that stamp.
......Michael Potter would presumably say (judging by his 2004 “Set Theory and its Philosophy,” page 22) that while that collection would have one member, that stamp would have a value, a weight, a design, but no member, but I think that it would only be misleading to say that it had one member. Our way of talking about a collection’s members is taken from the paradigm case of a plurality, and there the collection has members in the way that a composite object has component parts; and it is not exactly wrong to say of a simple object that it has only one (improper) part. (Similarly a particular photon has a particular energy, but nonetheless that photon is that energy; and a person both has and is his or her personality, his or her mind, etc.)
......Furthermore if our collector got rid of his last remaining stamp, then although he might have an empty stamp album, he would surely no longer have a stamp collection. And yet there is (supposed to be) an empty set, so a set is not just a collection of things—it is something more, but what? That something more is presumably like that album, but less specific; it is presumably something that all collections have as a matter of logic (rather than physics, or convention etc.).
......Now we have, in logic, classes of things (e.g. the class of men, the class of men now in this city, the class of classes), and we have empty classes, e.g. the class of aunts that are now in this room, and the class of ants that are now in this room. In one sense (intensional) those are two classes, with different modal properties (e.g. had I left the window open, ants might have got in), but in another sense (extensional) they are the same because they do not have different members. And that latter sense is the one we want because there is only one whole number 0. But surely the extensional class is just the collection of the class’s members (and we have already rejected collections), so the empty set is not a logical class. But I cannot think of anything else that sets could be, if not formal.

1 comment:

SamD said...

I first had to come to grips with set theory in my honours year when I read "St. Anselm's ontological argument succumbs to Russell's paradox", by Chris Viger. It's worth a read.

I had always accepted, so long as it was properly defined, that a set existed, even if it had no members. So I would agree that what logicians mean by a set is not so much like a collection. If I were to look it them in that way I'd perhaps see them as potential collections. Further than that I'm not sure what it means to say that an empty set exists.