Here’s a proof that any word in a language cannot be described within the same language.
Proof by Induction
Assume that each word takes at least one other word to define. Also assume that circular definition is not allowed. And finally, assume that a language can be constructed by adding word one by one.
Basis: Suppose a language consists of one word. Then trivially, this word cannot be described in terms of other words in the language. Thus the word cannot be described.
Inductive step: Suppose the statement is true for . Then add one more word T to this language, constructing a language with k + 1 words. Now suppose the statement is false; i.e. suppose that T can be described in terms of other words. But then T must be described in terms of at least one word in the original k words. Let us call such a word S. But S must be described in order to define T. However, by our inductive hypothesis, S cannot be described. Then T that cannot be described and we’ve reached a contradiction. Thus any word in a language consisting of k + 1 words cannot be described.
Hence basis and inductive steps have been proven. By mathematical induction, the statement is true. Q.E.D.