Skolem's Paradox

There are science fiction movies in which entire planets populated by alien civilizations appear. How do the movie makers manage to create an entire planet, civilization, history, involving perhaps billions of aliens over thousands of years without going waaaaay over budget? We all know the answer: they only create the part you see in the movie. Every street corner the camera pans by shows an intersection to another street that exists only up to the point you see it in the shot, no further. Alvin Goldman, a well-known epistemologist, talks about barn façades in a different, related context: when you see a barn façade driving in rural Vermont, you have good reason to believe there is a farm there. If you see the same thing on a Hollywood sound stage, you don't.

Set theory, for example, apparently describe a really really huge domain: a proper class of sets of many infinite sizes. There are more such sets, the theory tells us, than there are sentences in any language. If all we care about getting right is what we can say about the sets (in the language of first-order predicate logic), we can use the trick of the Hollywood special effects departments: we only need to actually have the sets we can manage to mention. Thus, we can create a countable model of the theory in which everything is true that is true of "the real thing." More is true: we can do that building a model out of the actual sets—we just keep the ones we need.

The trick is clear enough, though one needs a fair bit of mathematical work to show that it functions as advertised. But there seems to be a problem: Suppose we look inside one of Skolem's countable models of set theory, one of the set theory façla;ades, which are usually called Skolem hulls. There is a set which, in the hull, is said to be uncountable. Inside the hull, the following is true: there is no one to one correspondence between and the natural numbers.

Aside: the natural numbers in a Skolem hull may as well be taken to be the genuine natural numbers.

But, and this is the so-called paradox, the whole hull is countable, it only has countably many members, and so the "uncountable" has only countably many members. Thus, there is a one-to-one correspondence between and the natural numbers, despite the fact that the hull says there isn't. How can that happen? What is a one-to-one correspondence in the sense of set theory? It is itself just a certain set, a set of pairs, the things that "get corresponded to each other." There is a such a set of pairs that pairs off with the natural numbers, but that set of pairs is not in the hull. When it is said in the hull that there is no one-to-one correspondence, that means that there is no one-to-one correspondence in the hull. We get away with this by leaving lots of correspondences out, and that escapes notice because there are far more correspondences than ways to describe correspondences, and so we have no way to mention most of them.

Since the hull includes everything we can explicitly define (that was the trick), it follows that the correspondence between and the natural numbers is not definable. And that is one important aspect of why mathematics is so slippery: we often make use of the fact that functions or correspondences exist even when there is no way to explicitly set out a particular one.

--Main.ProfessorShaughanLavine - 24 Mar 2005