If we have abundant properties in our ontology, do we need to posit a second kind of entities, the sets?
Properties are kind of like sets. If P is a property, write x∈P if and only if x has P. A whole bunch of the Zermelo-Fraenkel axioms then are quite plausible. But not all. The most glaring failure is extensionality. The property of being human and the property of being a member of a globally dominant primate species have the same instances, but are not the same property.
We can get extensionality by a little trick and an axiom. Assume the following Axiom of Choice for Properties:
- If R is any symmetric and transitive relation, then there is a property P such that (a) if x has P, then x stands in R to itself, and (b) for all x if x stands in R to itself, there exists a unique y such that x stands in R to y and y has P.
To a first approximation, then, we can think of those entities that have S1 as sets. Then every set is a property, but not every property is a set. We certainly have extensionality, with the usual restriction to allow for urelements (i.e., extensionality only applies to sets). All the other axioms of Zermelo-Fraenkel with urelements minus Separation, Foundation and Choice are pretty plausibly true (they follow from plausible analogues for properties on an abundant view of properties). We get Choice for sets for free from (1).
Unfortunately, we cannot have Separation, however. For the property S1 is coextensive to some set U by our assumptions. And the members of U will just be the instances of S1, i.e., all the sets. And so we have a universal set, and of course a universal set plus Separation implies Comprehension, and hence the Russell Paradox.
So matters aren't so easy. The Axiom of Foundation is also not so clear. Might there not be a self-instancing property?
Thus the above simple approach gives us too many sets. But there is a solution to this problem, and this is simply to postulate the following second axiom about properties:
- There is a property S2 of properties such that (a) concreteness has S2, and (b) all the axioms of Zermelo-Fraenkel Set Theory with Urelements minus Extensionality are satisfied when we stipulate that (i) a set is anything that has S2 and (ii) A∈B if and only if A is an instance of B.
Now suppose that S1 is as before, and let S2 be any property satisfying (2). Then let S be the conjunction of S1 and S2. It is easy to see that if we take our sets to be those properties that have S, we will have all of Zermelo-Fraenkel with Choice and Urelements (ZFCU). Or at least so it seems to me—I haven't written out formal proofs, and maybe I need some further plausible assumptions about what abundant properties are like.
Of course, we cannot expect S1 and S2 to be unique. So there will be multiple candidates for sets. That's fine with me.
The big question is whether (1) and (2) are true. But if the theoretical utility of positing sets is a reason to think sets exist, then theoretical utility plus parsimony plus the reasons to believe in properties are a reason to think (1) and (2) are true.
I don't like the approach using (2). It's not elegant. A better move seems to be to define a separable property P as one for which separation holds: i.e., roughly, for any formula F(x), there is a property Q which is had by all and only the instances of P that satisfy F(x).
ReplyDeleteIt's plausible that a lot of the axioms of non-extensional set theory will be satisfied not only by taking sets to be properties but by taking sets to be separable properties. (In some cases, this is easy. For instance, the axiom of union for separable properties follows from the axiom of union for properties.) Getting the details of exactly what formal assumptions are needed will take some work, but to our second approximation we can take the sets to be all the instances of S1 that are separable. We will need some more assumptions to get Foundation (perhaps a further restriction on just which properties count as sets) and Infinity will probably be substantively stronger for separable properties than just for properties.
This is very cryptic as it's mainly a note to self.
A curious consequence of this approach is that sets might have their members non-essentially, and that something could be a set in the actual world without being a set at all in another world. For the choice of S1 may depend on contingent matters.
ReplyDeleteWe can maybe partly escape this by supposing that for any property P there is a coextensive property P' with some strong modal properties, such as that it's necessary that P' has exactly the instances it actually has. But I don't know how to do this exactly.