# First number, which can not be defined

What is the appropriate answer to this question?

What is the first natural number, which can not be defined with less than 15 words?

Explanations:
1. Natural means integers greater than 0.
2. Word here is any group of letters separated by spaces or punctuation. "a", "of", for example, are words too.
3. You are not allowed to use digits. All numbers in the definition must be written with words. 1234 takes 6 words, for example.

• I would like to suggest that this is in fact Berry's Paradox.
– user1994
Aug 3 '14 at 13:33

There is no such number. Like Ross mentioned in his answer, this is related to Richard's paradox.

At the heart of both this puzzle and Richard's paradox is (from the Wikipedia article)

the resolution of Richard's paradox is that there is no way to unambiguously determine exactly which English sentences are definitions of real numbers

The resolution of this puzzle is that although it sure seems like we are defining a real number, we are actually not defining one. For example, consider the following statement:

The first natural number that is greater than five and less than three.

This obviously does not describe a number as no matter which number you try one of the two requirements will always be left unsatisfied. Likewise, the statement

The first natural number, which can not be defined with fewer than 15 words.

has a requirement that cannot be met - it must not be able to be defined with fewer than 15 words, and yet we have done so. Thus, there is no such number. The only real difference between the two statements is in how obvious the unsatisfiable requirements are.

This is known as Richard's paradox. This is paradox because the phrase "the first natural number, which can not be defined with less than 15 words" has only 14 words and can be used as definition of a number.

• It took me a minute to understand this answer. Another way you could have said it is this - Because the phrase "the first natural number which can not be defined with fewer than fifteen words" only has 14 words in it, you have defined it in fewer than fifteen words and therefore no such number exists. This is a good example of a Richard's paradox. Aug 1 '14 at 4:18
• @RobWatts, you can't say that no such number exists, I believe. Aug 2 '14 at 4:56
• @klm123 My reply was too long for a comment, so I wrote it up as an answer. Aug 2 '14 at 5:34
• Small nitpick - this isn't actually Richard's paradox, as that is a specific paradox. It is closely related though. Aug 2 '14 at 5:43