Skolem's Paradox

Theorem
Let $\mathcal{L}$ be a countable first-order language, and let $T$ be an $\mathcal{L}$-theory which axiomatizes some version of set theory (for example, ZFC).

There is a countable model of $T$.

Why is this called a paradox?
This theorem is involved in an observation by Skolem that despite this theorem, within versions of set theory such as ZFC, statements can be proven that are interpreted as asserting the existence of uncountable sets. Since the theorem above guarantees countable models of ZFC, this presents a seemingly impossible state of affairs:


 * Theorem 1) A model of ZFC exists whose universe is countable.
 * Theorem 2) It is a theorem in ZFC that there exists an uncountable set.
 * Conclusion) There is a model of ZFC which is countable but has an uncountable subset.

This argument is flawed however, which Skolem understood. The problem with this argument is that it commits something of an equivocation fallacy. The resolution of this apparent problem lies in paying careful attention to what is actually being said.

What we mean when we say that we can prove a theorem in ZFC which says that there is an uncountable set is that there is a statement provable in ZFC such as
 * $\displaystyle \exists x \left(\neg\exists f \left(\mathrm{Function}(f) \wedge \mathrm{Bijection}(f) \wedge \mathrm{Domain}(f)=x \wedge \mathrm{Range}(f) = \mathbb{N}\right)\right)$

where several component formulas have been abbreviated with names.

We often interpret this formula in natural English language as "there exists a set $x$ such that there is no bijection between $x$ and $\mathbb{N}$".

However, this suppresses an important point which is highlighted when it is more accurately read as "there is some object $x$ in the universe for which no object $f$ exists in the universe such that certain formulas involving x and f hold".

Perhaps most important to notice is that the quantifiers involved in this statement range over the universe of discourse, and models satisfy this statement so long as it is true when interpreted with the variables ranging over the models universe.

So, as far as models are concerned, all the ZFC statement above really says is that for any particular model $\mathcal{M}$ of ZFC, there must be an object $x$ in the universe of $\mathcal{M}$ such that there does not exist an object $f$ in the universe of $\mathcal{M}$ which is interpreted as a bijection between $X$ and $\mathbb{N}$.

In short, Theorem 2 in the fallacious argument above only tells us that a model's universe must not contain something it interprets as a bijective function with certain properties; it does not tell us that actual bijections can't exist outside of the universe of the model. This can be summarized with a quick informal statement: being "countable" from the perspective of models of first-order formalizations of ZFC is not the exact same thing as being "countable" from a perspective outside of these models.

Proof
This is a straightforward application of the downward Löwenheim-Skolem Theorem.