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Skolem’s Paradox [Quiz]

Genady
in Mathematics

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There is a theorem in logic that says that given a countable formal language, any consistent formal theory in this language has a countable model.

What is paradoxical about this statement? For example, the language of the axioms of Zermelo-Fraenkel Set Theory ZFC only consists of the membership relation ∈. Hence, if ZFC is consistent, it has a countable model. However, it is easy to prove from the axiom system ZFC that there exist uncountable sets.

How is it not a contradiction?

Edited by Genady

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