A simple lesson from Skolem's "paradox"


Posted on July 2, 2021

Here is a lesson that I learned the hard way.

Skolem’s paradox is the apparent mismatch between the following:

  1. Within \(\mathbf{ZFC}\), we may show that nonenumerable sets exist, such as \(\mathbb{R}\). So, there must be uncountably many sets in \(\mathbf{ZFC}\). Let’s call this the “internal view”, as it is a theorem within \(\mathbf{ZFC}\).

  2. \(\mathbf{ZFC}\) is a first order theory. Therefore, by the Löwenheim-Skolem theorem, it has a model \(\mathfrak{M}\) with an enumerable domain, i.e. there are only countably many sets in \(\mathbf{ZFC}\). Let’s call this the “external view” as it is a theorem about \(\mathbf{ZFC}\).

So the paradox is the seeming contradiction between the internal view and external view of \(\mathbf{ZFC}\). At first glance, I took this to mean something was deeply wrong with first-order logic or \(\mathbf{ZFC}\) or both. But upon further reading, it became clear that this paradox is a fake one, and I fell for it hook, line and sinker.

To discredit this faux paradox, I will point out and resolve the ambiguity of both the internal and external views.

Let’s start by tackling the internal view. \(\mathbf{ZFC}\) is a formal theory, a collection of sentences composed of symbols. It is not a model. Hence, the phrase “there must be uncountably many sets in \(\mathbf{ZFC}\)” is, by itself, meaningless. To read a meaning out of the phrase, you must implicitly assume an interpretation/model of the symbols in \(\mathbf{ZFC}\), a personal bias of what the symbols mean to you, which is that \(\mathbf{ZFC}\) is a theory of pure sets. Let’s call this implicit interpretation \(\mathfrak{S}\). In this interpretation, the internal view holds.

Now in the second point, we invoked the Löwenheim-Skolem theorem to explicitly put forward a different, maybe even esoteric interpretation \(\mathfrak{M}\) with an enumerable domain. Calling the objects in this domain “sets”, the external view must hold of \(\mathfrak{M}\).

But as the internal and external views are statements about different models, they are therefore mutually independent. No paradox.

The real problem is this: we constructed \(\mathbf{ZFC}\) to precisely capture just the notion of pure sets (which has to be nonenumerable) yet the Löwenheim-Skolem theorem means it also captures additional unwanted notions. We cannot change the first order theory we use as the Löwenheim-Skolem theorem applies just as well to any theory. So first order logic must be incapable of exactly capturing the notion of pure sets. But this is a different problem altogether for another day.

TL;DR: The lesson is that formal theories are just symbols - don’t read too much meaning into what the symbols mean.