Topics Mathematics

Question I’ve run into a problem in philosophy recently that I do not completely appreciate. Certain sets are said to be “too big” to be sets. In Lewis’ Modal Realism, the set of all possible worlds is said to be one such set. These are sets whose memberships is composed of infinite individuals of a robust cardinality. I (purportedly) understand that not all infinities are equal. But I don’t quite see why there can be a set of continuum many objects, but not a set of certain larger infinities. Am I misunderstanding what it is to have “too big” a set?

Accepted:
November 27, 2006

Comments

When a set theorist says that such and such collection is "too big" to be a set, what he typically means is that if that collection were taken to be a set a contradiction would arise. The collection of all sets is such a collection. If we assume it's a set then, applying the argument that generates Russell's Paradox, we arrive at a contradiction. And so we conclude that we were wrong to assume that the collection of all sets is a set. As set theorists put it, it's a proper class, not a set. Is there any way of telling whether a collection is "too big", i.e., a proper class, or whether it's a set, besides seeing whether the assumption that it's a set leads to a contradiction? No. So really, "too big" is just a colorful way of saying "leads to a contradiction if assumed to be a set".

As Alex says, in Lewis's case, he's really pointing towards an idea familiar from the philosophy of set theory. Not all "collections" of objects can form sets: The assumption that they do leads to contradiction. (Of course, we need some logical assumptions to get that contradiction, and these could be denied. And one might also not think contradictions are all that bad. But let's not go there now.) And given the standard axioms of set-theory, we can prove that there is no set containing every object and, moreover, that there is no set that can be put in 1-1 correspondence with all the objects there are. But the standard concept of set, as embodied in Zermelo-Fraenkel set-theory, is not in any way motivated by the idea that a set cannot be "too big". It is based upon a very different idea, called the "iterative"conception of set, though there is some question about whether the Axiom ofReplacement (which was Fraenkel's distinctive contribution) is really motivated by the iterative conception or ratherneeds some other motivation, and the "not too big" idea is anatural one in that case. See George Boolos's paper "Iteration Again"for more on this issue.

There are, however, some conceptions of what a set is that take the "too big" idea seriously. These are so-called "limitation of size" conceptions. Georg Cantor is often credited with such a conception. According to Cantor, if you have some objects, these objects may form a set. But they may also constitute what he called an "inconsistent multiplicity" that is, as he put it, be "absolutely infinite" and so too big to form a set. This was very important to Cantor, who wrote several letters to the Pope explaining that his theory of infinity was in no way intended to have theological implications and that, indeed, the idea of "absolute infinity" somehow reflected the idea that God is ultimately unknowable by finite minds. So far as anyone can tell, it never occurred to the Pope to be worried about this. Perhaps what's really odd about the story is that Cantor wasn't even Catholic.

One way to formalize a limitation of size conception is this. We'll assume we have a logic in place that allows us to talk about "collections" of objects. (This could be second-order logic or a logic of plurals.) Say that a collection of objects is big if it can be put in 1-1 correspondence with the collection of all the objects there are. Now we take the following two axioms.

Set existence: Any collection that is not too big forms a set whose members are exactly the things in that collection.
Extensionality: Sets X and Y are identical if, and only if, they have exactly the same members.

It can be shown that this theory is consistent and, moreover, that it proves many of the usual axioms of set theory (given some appropriate definitions). It does not, however, prove the so-called Power Set Axiom, which says that, if you have a set X, then there is a set Y that contains as members all and only the subsets of X. That really shouldn't follow: How do we know that, if X isn't too big, then the set of all its subsets isn't too big either? Cantor proved that the latter set has to be bigger than the former, so, well, we don't know that. This makes the theory quite weak, as compared to Zermelo-Fraenkel set theory or even with respect to Zermelo set theory. It can be shown, in particular, that it is exactly as strong as the theory of (real) analysis. (This particular theory is due to George Boolos: See his paper "Saving Frege From Contradiction".)

Another idea, more recent, is due to Harvey Friedman and John Burgess. Here we think of "too big" as meaning "too big to be comprehended"---developing Cantor's theological take on limitation of size. What that means, intuitively, is that you can never tell when you're really thinking of such a thing---say, the collection of all sets---or are instead thinking of something smaller. There is a precise was to explain this formally, in terms of a technical notion called "relativization". But what's really interesting is that the resulting theory---same the preceding one, except that axiom (1) now talks about this notion of bigness---is very, very strong. For an exposition of this theory, see Burgess's book Fixing Frege.