Topics Mathematics

Question Consider a first-order axiomization of ZFC. The quantifiers range over all the sets. However, we can prove that (in ZFC) there is no set which contains all sets. Soooo.........how can we make a _model_ for ZFC? The first thing you do when you make a model for a set of axioms is specify a domain, which is a set of things which the quantifiers range over......this seems to be exactly what you can't do with ZFC. So what am I missing?

Accepted:
December 20, 2007

Comments

You are right that the first thing you do when you make a model for set theory is to specify a domain. You also have to specify an interpretation for the "is an element of" symbol. For example, you might specify that the domain is to be the set of natural numbers, and the symbol for "is an element of" is to be interpreted to mean "is less than". Of course, under this interpretation most of the axioms of set theory would come out false, so although this is a possible interpretation for the language of set theory, it is not a model of ZFC. But if ZFC is consistent, then it has a model whose domain is the set of natural numbers. In this model, the "is an element of" symbol would be interpreted as some relation on the natural numbers. (The existence of such a model follows from the Lowenheim-Skolem theorem, which says that if ZFC is consistent, then it has a countable model.)

Now, you might object that this model is not the intended model, because our intention is that the variables in the language of set theory should stand for sets, and the "is an element of" symbol should be interpreted to mean "is an element of". So you might want to restrict attention to models in which all elements of the domain are sets, and the "is an element of" symbol has its intended meaning. Let us call such models "standard set models". Such models can exist. For example, the existence of inaccessible cardinals implies the existence of standard set models of ZFC. Although the existence of inaccessible cardinals is not provable in ZFC, most set theorists think their existence is consistent with ZFC. (A bit more detail, for those with some knowledge of ZFC: If kappa is an inaccessible cardinal, then the set of all sets whose rank is less than kappa is a standard set model for ZFC.)

Standard set models are closer to our intended interpretation of the language of set theory. But you still might object that even a standard set model does not interpret the language in the intended way. Although the domain of a standard set model contains sets, it doesn't contain all sets, because, as you observe, in ZFC there is no set that contains all sets. For example, if X is the domain of a standard set model of ZFC, then X is a set, but X doesn't contain itself as an element. If we speak in a somewhat colorful way and say that the elements of X are the things that model "thinks" are sets, then we could say that the model doesn't think X is a set, even though it really is one.

The only model that seems to completely agree with our intentions about how the language of set theory should be interpreted is the one in which the domain is the collection of all sets, and the "is an element of" symbol is interpreted to mean "is an element of". And in this interpretation, the domain is not a set, but is rather what is called a proper class. Set theorists do sometimes study such proper class models. The study of proper class models requires care. For example, in the language of set theory variables only stand for sets, so you can't say something like "for every x that is the domain of a proper class model of ZFC, ..." But it is still possible to prove some interesting theorems about proper class models. For example, Godel defined what are called the constructible sets, and he proved that the collection of all constructible sets is a proper class that is the domain of a proper class model of ZFC.

This kind of concern has had a good deal of influence on research in logic over the last several decades. It was, for example, a major force behind Boolos's work on plural quantification.

More recently, there has been an explosion of research on what is called "absolutely universal" quantification: quantification over absolutely everything, including all the sets there might be. As you note, there is no "model" of such discourse, in the usual sense; that is, the "intended interpretation" of such discourse cannot be a model, in the usual first-order sense. As Dan noted, one can talk about proper class models, but there is another line of inquiry, deriving from Boolos. One way to develop this approach is to take the domain of the interpretation to be a `plurality', so that the quantifiers range over the sets---not the set ofall sets, or a class of all sets, but simply over the sets, whatever sets there may be. The details have to be worked out here, but it can be done, and in a reasonable way, too, as Tim Williamson and Agustin Rayo have shown.