TITLE: Proof that $\mathbb N $ is finite
QUESTION [15 upvotes]: Obviously this is a false proof.  It relies on Berry's paradox.
Assume that $\mathbb{N}$ is infinite.  Since there are only finitely many words in the English language, there are only finitely many numbers which can be described unambiguously in less than 15 words.  Let $n$ be the smallest number which can't.
Then $n$ can be described as "the smallest number which can be described unambiguously in less than 15 words".  Contradiction.
I know nothing of mathematical logic, but looking in a few books has told me that the problem here lies in the definition $n$ := "smallest number which can't be described unambiguously in less than 15 words".  If this isn't a valid definition, then what exactly is a valid definition?

REPLY [13 votes]: Essentially what your argument shows here is that "described unambiguously in less than $N$ words" is not itself an unambiguous description -- we can plainly see ambiguity arise in the form of the paradox. This defuses the paradox, because now description itself is not among those we quantify over.
It is of course easy to accuse natural-language phrases of being ambiguous, but the same problem carries over if we attempt to formalize the argument. For example, we could ask for the least natural number $n$ such that for every first-order formula $\phi(x)$ containing less than $20,000$ symbols in the language of basic arithmetic, $\forall x.(\phi(x)\leftrightarrow x=\bar n)$ is false in $\mathbb N$.
The wall we then hit is this: Even though we can represent the formulas $\phi(x)$ themselves inside arithmetic using Gödel numbers, there is no arithmetic formula that expresses the property of being the Gödel number of a true formula. So the number asked for in the previous paragraph is indeed not described by any arithmetic formula.
We can define arithmetic truth if we allow formulas in a stronger language, such as set theory. But that still doesn't produce a paradox, because the language of set theory cannot express set-theoretic truth.