The Skolem paradox is a result of the "downwards" Skolem-Löwenheim theorem, which says that any countable system of axioms has a countable model. Axiomizations of set theory (such as Zermelo-Fraenkel) thus have theorems which appear to speak about "uncountable" sets, but can be satisfied by only a (much smaller) countably infinite number of objects.

For example, the power set **P(N)** of the natural numbers is strictly larger than the set of natural numbers **N**. But the S-L theorem says that we can use a set the same size as **N** as a model for *all* sets, including **N**, **P(N)**, and the infinite chain of larger and larger power sets. This seems to contradict the fact that not all infinities are equal.

Certainly by the *normal* interpretation of the words used in the axioms requires an uncountably infinite model.
But the paradox arises from confusing this "English" interpretation with the one imposed by a particular model.
If we adopt a particular interpretation (and thus a model), we can't immediately take it back and demand that this mapping fit different criteria.

For example, we tend to think of a particular meaning for "`straight line`" in Euclidean geometry, but the first four postulates can be satisfied by entities which are curved, giving rise to non-euclidean geometry. Just as there's no real paradox in non-straight "`straight lines`", there's no need to be concerned about
countable "`uncountable sets`". Applying the Skolem theorem to the real numbers gives a nonstandard interpretation in which "`real numbers`" are one-to-one with the natural numbers, from our normal perspective,
but not "`one-to-one`" with the "`natural numbers`" inside the model.

The paradox seems to say more about the failures of formal systems to capture human mathematical thinking than about any weakness in set theory.