This is really an interesting question, though I do not know how to word it in a mathematical way. I am glad if one can help me to reword it mathematically.
A friend of mine comes up with this theorem:
Theorem. Any positive integer can be uniquely determined by a sentence of less than 20 words describing it.
Example. 479001600 can be determined by the sentence "Twelf factorial." 60000000000 can be specified by the sentence "Six followed by ten zeros". Each of the two sentence contains less than 20 words.
Proof. Let's prove by contradiction. Suppose the theorem is not true, the set $C$ of counterexamples is nonempty. Since $\mathbb{Z}_+$ is well ordered, $C$ has a smallest element $x$. But $x$ can be specified by the sentence "The smallest positive integer that canNOT be uniquely determined by a sentence of less than twenty words describing it", which is a contradiction.
Well the theorem is clearly false because the set $\mathbb{Z}_+$ is infinite while the set of sentences of less that 20 words is finite.
How can one reword this paradox mathematically, and explain why this happens?
ps: It appears to me that this is a self reference of the language. ps: I have some experience working with formal language, but I don't know how to explain this paradox.
I am not completely sure how to tag this question, so please help me to correct wrong tagging.