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.

## Also see

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

## Source of Name

This entry was named for George Godfrey Berry.