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?