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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.