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Every Natural Number can be Unambiguously
Described in Fourteen Words or Less
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 you might find in a
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
- 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.
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