The claim is that any natural number can be completely and unambiguously identified in fourteen words or less. Here a "word" means an ordinary English word, such as one you might find in a dictionary.

You know this can't be true. After all, there are only finitely many words in the English language, so there are only finitely many sentences that can be built using fourteen words or less. So it can't possibly be true that every natural number can be unambiguously described by such a sentence. After all, there are infinitely many natural numbers, and only finitely many such sentences!

And yet, here's a supposed "proof" of that claim. Can you figure out what's wrong with it?

The Proof:

Suppose there is some natural number which cannot be unambiguously described in fourteen words or less.

Then there must be a smallest such number. Let's call it n.

But now n is "the smallest natural number that cannot be unambiguously described in fourteen words or less".

This is a complete and unambiguous description of n in fourteen words, contradicting the fact that n was supposed not to have such a description!

Since the assumption (step 1) of the existence of a natural number that cannot be unambiguously described in fourteen words or less led to a contradiction, it must be an incorrect assumption.

Therefore, all natural numbers can be unambiguously described in fourteen words or less!

You know this can't be true. After all, there are only finitely many words in the English language, so there are only finitely many sentences that can be built using fourteen words or less. So it can't possibly be true that every natural number can be unambiguously described by such a sentence. After all, there are infinitely many natural numbers, and only finitely many such sentences!

And yet, here's a supposed "proof" of that claim. Can you figure out what's wrong with it?

The Proof:

n.nis "the smallest natural number that cannot be unambiguously described in fourteen words or less".nin fourteen words, contradicting the fact thatnwas supposed not to have such a description!## Share this post

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