One of the best papers at last year’s “BSPC”:http://www.ac.wwu.edu/%7Earistos/BSPC6/BSPC6.html (link to this year’s CFP) was Gabriel Uzquiano’s “The Price of Universality”:http://www.ling.rochester.edu/~uzquiano/PriceofUniversality.pdf. I think it was one of the better papers I heard at any of the (many) conferences I went to this year, the main rival being Tim Williamson’s “Must Do Better”:http://www.philosophy.ox.ac.uk/faculty/members/docs/Must%20Do%20Better.pdf, which certainly didn’t have as much _content_.
The core of Gabriel’s paper is a paradox he and Agustin Rayo have worked on. It seems difficult to believe the following four claims.
* Universalism about composition – any things have a unique fusion
* ZFC is the correct axiomatisation of set theory
* No gunk (of a particularly odd variety)
* Maximality – There is a 1-1 map from the entire universe into the pure sets.
Gabriel uses ‘no gunk’ to get the contradiction out, Daniel Nolan pointed out at Bellingham last year that this could be considerably weakened, since the kind of gunk that would be needed to avoid paradox would not be the kind of gunk gunk-lovers normally love. I’ll leave that debate and just work with the no gunk version of the paradox.
Maximality looks rather strange, though as Gabriel argues in the paper there are good reasons from the metaphysics of set theory to believe it, and I don’t want to reject it either.
So the problem is a clash between ZFC and universalism. The problem is that any model for ZFC requires a universe whose size is inaccessible. No gunk plus universalism requires that the size of the universe be 2n, where _n_ is the number of atoms in the universe, and hence be accessible. Contradiction.
We could try a few things to block the contradiction. We could note first that this talk about sizes of the universe is odd because we normally talk about sizes of sets, and there is no set of all things. Gabriel argues that won’t work because we can rephrase everything in talk of second-order logic with plural quantification, and restate the paradox that way. Daniel’s comments at Bellingham questioned just this point, and I don’t know enough about the dispute by this stage to safely take sides. But there do _seem_ to be a lot of benefits to allowing in the kind of plural talk that Boolos, Lewis and others used, so I’m going to (a little tentatively) take Gabriel’s side here. And in doing so I conclude there’s a real problem, one of ZFC and universalism has to be given up. Bearing in mind Lewis’s admonition against questioning what mathematicians say about mathematics, I’m going to keep ZFC and give up universalism.
If we go down this road, we have to decide how much to qualify universalism. Gabriel thinks we should say that the concepts of part and whole don’t apply to sets. So not only is no set part of another thing, no set is part of itself. That seems rather radical – I think keeping _part of_ as reflexive is a virtue. Rather than take this radical path, I prefer a minimally qualified mereology. In particular I favour the following view.
bq. Any sets have a fusion iff they have a union. If they have a union that union is their fusion. If some things include some sets, they have a fusion iff those sets have a union. In that case, the fusion is the fusion of the non-sets fused with the union of the sets.
I don’t have much of an argument for that view, save that it is the smallest qualification on universalism that avoids the paradox. It preserves all the usual axioms of mereology, since any two things, indeed any number of things, still have a fusion. And it does without proper classes, which seem like good things to do without. And it is immune to vagueness arguments, though Daniel has argued that those arguments are no good anyway.
But even without an argument, I have a worry. The worry is that maybe we should qualify further. Maybe we should start with my view, and add the view that no set can be fused with a non-set. That would mean giving up the finite axioms of mereology, since there would be two things, me and my singleton, that don’t have a fusion. But it would have an advantage, namely that we could say that for any set s and anything x, s is a part of x iff s is a subset of x. That is, we could preserve in a very strong way the idea that subsethood just is parthood defined over the sets. I favour keeping finite mereology over keeping the strong connection between subsethood and parthood. But as you may have noticed, I’m long on preferences and short on arguments around here.
The main points of this post, other than finally getting around to recording what I for now believe, are to encourage you to read “Gabriel’s paper”:http://www.ling.rochester.edu/~uzquiano/PriceofUniversality.pdf and to note what high quality papers are sent to the “BSPC”:http://www.ac.wwu.edu/%7Earistos/BSPC6/BSPC6.html, and to recommend that readers send similarly high-quality papers along to them this year.
