Missives from a fly bottle
barang dot sg
Last revised 31 August 2017
23. Every natural number can be unambiguously described in fourteen words or less

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 number N. Then N is the smallest natural number that cannot be unambiguously described in fourteen words or less, which is a complete and unambiguous fourteen-word description of N. Contradiction.