Every Natural Number 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 flaw is so subtle that it's not simply a matter of an obvious mathematical error in one of the steps. So we won't present this proof in the same way as the other ones in this collection. Rather than choosing a particular step and having the computer tell you if you're right or wrong, just try your best to figure out where the fallacy lies. When you've thought about it all you can, look at the A Discussion of the Fallacy section to see if you were correct or not.

The Fallacious 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!

When you have thought about this all you can and want to see a solution, go to the section A Discussion of the Fallacy.

This page last updated: May 26, 1998

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

