Imagine that you are an extremely talented, and extremely ambitious, shepherd, and an equally talented and equally ambitious carpenter. You decide that you want to explore what enclosures, or ‘fences‘, you can build, and which groups of objects, or ‘flocks‘, you can shepherd around so that they are collected together inside one of these fences.
As you build fence after fence, and form flock after flock, you begin reflecting on how your fences, and the flocks enclosed within them, work. Three things about building fences stand out:
First, you discover that you can not only collect together everyday objects to form flocks by building fences around them, but you can also form flocks of flocks by building large fences around smaller fences. For example, if you have built a fence around St Paul Minnesota, and you have built another fence around Minneapolis, Minnesota, then you can build a third larger fence that encircles both the fence around Minneapolis and the fence around St Paul, and a new flock is formed that contains both of the original city-flocks.
Second, you discover that you can subdivide flocks into smaller flocks. For example, if you have built a fence around Minneapolis, then you can build a smaller fence separating South Minneapolis from the rest of Minneapolis. And in fact you can do this for any flock and any sub-collection of objects in the flock: If X is some collection of objects contained in a flock you have already formed by building a fence, you can build a second fence inside the confines of the first fence that separates the objects in X from the other objects that are in the flock but not in X, to form a new, smaller flock that contains exactly the objects in X. You can even do this when X is empty, by building a fence so small that it contains nothing at all within its borders.
Third, you discover that fences don’t have a privileged side. Of course, sometimes fences look different on one side than another. But this is only an aesthetic concern–from the perspective of using fences to separate objects into flocks, there is no difference between one side of a fence and the other. As a result, if you build a fence that separates some collection of objects X from all other objects (the non-Xs) and forms the Xs into a flock, then that same fence also collects together the non-Xs and forms them into a flock.
Reflecting further on these three discoveries, however, you suddenly discover something disconcerting: building fences is impossible!
Occasionally you have doubts about this third discovery. But anytime you do, you imagine a fence built along the equator. You then ask yourself: would such a fence be a fence around all of the objects in the Northern Hemisphere, or a fence around all of the objects in the Southern Hemisphere? The only reasonable answer is that such a fence would be both, and what holds for the equator-fence should hold for all fences whatsoever.
Reflecting further on these three discoveries, however, you suddenly discover something disconcerting: building fences is impossible! Imagine that you build a fence–any fence–enclosing some collection of objects X. Then, by the second discovery, you could build a second fence that collected all of the objects in the first flock that are not identical to themselves–since there are no such objects, this would be a small fence forming a flock that had no objects whatsoever in it (what we might call the empty flock). But by the third discovery, if this fence separates all of the objects in the empty flock from every other object–that is, all objects whatsoever–then this same fence fences in a second flock that contains every object whatsoever (what we might call the universal flock). According to the first discovery, flocks are themselves objects that can be collected together and used to form other flocks. In particular, if the universal flock collects together all objects whatsoever, then the universal flock is in fact one of the objects that is contained in the universal flock. Thus, some flocks are in fact contained in themselves! Hence, by the second discovery, we can build a fence around exactly those objects contained in the universal flock that are themselves flocks that are not contained in themselves. But now we need only ask: Is this final flock contained in itself or not? By definition, it is if and only if it isn’t, and we are faced with a contradiction.
Now, at this point we could argue about whether it is the principles of carpentry, or the principles of shepherding, or both that are to blame. But this would be silly. The puzzle just described is a variant of a familiar set-theoretic paradox that has nothing to do with either carpentry or shepherding: The Russell paradox. We have formulated the puzzle a bit differently than is normally done, however. Instead of merely laying down a comprehension principle that states, loosely put, that for any condition C there is a set that contains exactly the objects that satisfy C, and then constructing the contradictory Russell set from there, we instead arrived at the paradox via combining two independently plausible set-theoretic principles:
- Given any set S, and any subcollection of objects X all of which are members of S, there exists a set that contains exactly those objects in X.
- Given any set S, there exists a set that contains exactly those objects that are not in S.
The first principle is an informal version of the Axiom of Separation, which is one of the standard axioms of the widely studied and widely accepted theory of sets known as Zermelo Fraenkel Set Theory with Choice (or ZFC). The second principle is the Axiom of Complement, which is not an axiom or theorem of ZFC (since if it were we could derive a version of the paradox above). But it is an axiom of other, alternative set theories, such as W.V.O. Quine’s New Foundations (NF).
Now, obviously we can’t have both Separation and Complement as axioms of our set theory. Most of us trained in mainstream contemporary mathematics have been taught that there is something natural and almost inevitable about rejecting Complement and retaining Separation in the face of the Russell paradox. But that seems wrong to me. As I have tried to show with the fence-building story told above, both of these axioms have a rather strong appeal grounded on basic intuitions one might have about collecting objects. Perhaps we–that is, the philosophical and mathematical community–have been too quick to opt for Separation (and hence ZFC). Separation does seem intuitively obvious, but then so does Complement–or at least it does to me.
So what exactly are we to do?
Featured image credit: ‘Sheep’, by Hans. Public domain via Pixabay.