# Berry Paradox

## Paradox

Every number can be defined by a sentence in natural language.

For the purpose of this argument, let that natural language be English.

It is assumed without proof that English has a finite number of words.

Let $n$ be an integer such that $n \ge 15$.

Then the cardinality of the set of integers that can be defined in no more than $n$ words is finite.

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?

## Resolution

The expression

- 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**.

## Notes

This paradox is related to Gödel's Incompleteness Theorems, specifically, Gödel numbers.

## Source of Name

This entry was named for George Godfrey Berry.

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 1.2$: Sets: Exercise $7$

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $111,777$

- Weisstein, Eric W. "Berry Paradox." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/BerryParadox.html