You know this can’t be true.

(Here a ‘word’ means an ordinary English word, found in a dictionary.)

After all, there are only finitely many words, so there are only finitely many sentences constructible in fourteen words or less. But there are infinitely many natural numbers! So it can’t be true that every natural number can be unambiguously described in fourteen words or less. Yet here’s a “proof” of the claim. What’s wrong with it?

Suppose the claim is false. Then at least one natural number cannot be unambiguously described in fourteen words or less. Call the smallest such numberN. ThenNis the smallest natural number that cannot be unambiguously described in fourteen words or less, which is a complete and unambiguous fourteen-word description ofN. Contradiction.

From math.toronto.edu

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