Every number can be defined by a sentence in natural language.
Let $n$ be an integer such that $n \ge 15$.
Consider the integer that is defined as:
- the smallest integer which cannot be defined by a sentence of at most fifteen words.
Let this number be $N$.
That is $N$ cannot be defined by a sentence of at most fifteen words.
But that very sentence itself has fifteen words.
So $N$ has been demonstrated to be definable in a fifteen-word sentence.
So: can it or can't it?
- the smallest integer which cannot be defined by a sentence of at most fifteen words
is self-contradictory, as any integer it defines can be defined in at most fifteen words.
The problem arises because the definition itself contains the word "defined".
To formalize a statement like this, it would be necessary to first define the word "defined".
Also known as
Some sources cite this as Berry's paradox.
Source of Name
This entry was named for George Godfrey Berry.
- 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica: Volume 1
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets: Exercise $7$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $111,777$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Berry's paradox