# A word cannot be described within the same language

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 $$k\ge 1$$.
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.