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Instead of using abstract logic, model theory, and axiomatic set theory, I will only use standard English language and a little bit of naive set theory at a high-school level.

By the well known diagonal trick, Cantor has shown that the set or real numbers is not countable. But he has proven it by using only English language (actually German, but that's irrelevant here) enlarged with a few basic mathematical symbols. Such a language (or more precisely, the set of all sentences expressible in that language) is certainly countable. So by expressing set theory in a countable language, and therefore by using only countable sets, he has proven that there is an uncountable set. But that's a paradox, for how can a smaller countable set prove the existence of a larger uncountable set?

Is the paradox above at least a good analogue of the Skolem paradox? If not, then why not? Can you make a better simple analogue of Skolem paradox?