Analytic constraints
Daniel Nolan and I once suggested that talk about sets should be analyzed as talk about possibilia. For simplicity, assume we somehow simply replace quantification over sets by quantification over possible objects in our analysis. This appears to put a strong constraint on modal space: there must be as many possible objects as there are sets.
But does it really? "There are as many possible objects as there are sets." By our analysis, this reduces to, "there are as many possible objects as there are possible objects". Which is no constraint at all!
In some form or other, the problem is very common. For instance, the possible worlds analysis of belief also seems to put strong constraints on what worlds there are: whenever one can believe that things are such-and-such, there must be a world where they are such-and such. Similarly for the possible worlds analysis of modal operators, and for possible worlds semantics. And similarly, though much more indirectly, for Lewis's principle of recombination, which he offers after noticing that the other constraints are empty (Plurality, sec. 1.8): once we plug in Lewis's analysis of "intrinsic" and "duplicate", the recombination principle, too, becomes virtually empty. Or suppose one analyzes properties as sets of instances, or belief content in terms of causal relations, or laws in terms of regularities. In each case, it seems that the analysis puts strong constraints on the analysans, even though this constraint turns out to be empty on the assumption that the analysis is correct.
The problem is distantly related to the 'paradox of analysis': how can an analysis not be trivial? If our proposed analysis was trivial, if everyone immediately agreed that by "set" of course they mean "possible object", the constraint would not appear strong. It would be like the constraint that there must be as many unmarried men as there are bachelors.
However, not all analytic truths are trivial, in the sense of immediately obvious. It is not immediately obvious that there is no decision procedure for first-order logic, yet this is a logical (and hence analytical) truth. Our proposed analysis of set talk is meant to be non-trivially analytic. It is meant to be analytic though; it is not meant to be a stipulative redefinition of "set", nor is it meant to be an empirical hypothesis about what sets will turn out to be once we catch some of them and investigate them in the lab. If the analysis is analytic, the apparent constraint -- that there are as many possible objects as there are sets -- remains empty. Non-trivially empty, but still empty. Which still seems wrong.
The constraint appears non-empty because it links concepts from two apparently different, and individually rich, theoretical frameworks: mathematics and modal metaphysics. Suppose we had no clear prior grasp of the concept /possible object/. We could then introduce the concept by a theoretical role: the possible objects are things X such that 1) whenever it is conceivable that something is such-and-such, there is an X that is such-and-such-prime (e.g. that encodes being such-and-such), ..., n) whenever there is a set that is such-and-such, there is an X that is such-and-such-prime. This list of roles is not satisfied by any old collection. In particular, to satisfy the role, there must be as many of the candidate objects as there are sets. This is our constraint, and it is not empty, as witness the fact that it excludes the coins in my pocket. Given that something at all satisfies the constraint, it is indeed empty that the possible objects, defined by the above role, satisfy the constraint; if any things don't satisfy the constraint, they don't play the role, and therefore can't be the possible objects.
A constraint is a property; to satisfy the constraint is to have the property. A constraint is empty if everything has the property. "There are as many possible objects as there are sets" doesn't strictly speaking express a constraint, because it expresses a proposition, not a property. Or maybe it does express a constraint, because propositions are properties of worlds. But this constraint corresponding to the expressed proposition is really empty. To express the intended constraint, we have to express the right property: "there are as many of the X as there are sets". Following the above definition of "possible objects", the claim that the possible objects (if they exist) satisfy this constraint is still empty. But that is as it should be. If possible objects are defined as the things that have a certain property, nothing much is said by the claim that they have this property.
The true situation is more complicated because the reductive analysis of set theory isn't one of the clauses by which we've learned what possible objects are. Nor is it one of the clauses by which we've learned what sets are. We have an independent grasp of the two notions.
What motivates our reductive proposal is that the roles that actually define the two notions are incomplete, and compatible: all we know about sets is that they satisfy the second-order Peano Axioms; and all we know about possible objects entails that, suitably interpreted, they satisfy the second-order Peano Axioms; but if the X are defined as things that do such-and-such, and the Y actually do such-and-such, why not identify the X with the Y?
It is not immediately obvious that the possible objects satisfy the Peano Axioms (under a suitable interpretation), mainly because it is not immediately obvious that there are proper-class many of them. Fortunately, there are good reasons to believe that there are proper-class many ways a thing might conceivably be. Hence if it is definitional of possible objects that whenever it is conceivable that something is a certain way, there is a possible object that is that way, then the possible objects will be enough. That is, if we reduce the above definition of possible objects to its first clause -- "the possible objects are things X such that whenever it is conceivable that something is a certain way, there is an X that is that way" -- this will ensure that whatever qualifies as possible objects will satisfy the constraint. As before, the constraint itself is strong; that the possible objects satisfy it, is empty; though this time, it is non-obviously empty.
What if there were no good reasons for believing that there are proper-class many possible objects, nor any reasons to disbelieve it? Then our analysis would become slightly revisionary. We have a shopping list of roles associated with "possible objects", and another one associated with "sets". Our proposal would be that we should add another item to the list for "possible objects", because that will allow for a smooth reduction of mathematics and thereby serve ideological and ontological simplicity. It would be like Fraenkel's proposal to add the axiom of replacement to the list for "sets" because it allows us to construct some nice things out of sets (like the von Neumann ordinals) that would otherwise be impossible.