Paradoxical Proof?

[BAndrew]

My friend challenged me to solve the following problem*. I seriously haven't found the solution yet, but I would like to hear your ideas (and no I don't want to steal your ideas and go to play the smart guy or something. That would be pointless.). In my opinion it is a great puzzle which can astonish even experienced solvers. Please, if you think you have an answer put it into spoilers.

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Below we will prove that every natural number can be described in fourteen words or less. Natural numbers are called integers that are greater than 0. By words we mean any English (or other language) words contained in any dictionary and which must form a phrase with a meaning. For example, the phrase "the natural number between three and five" specifies the number 4.
The proposal is obviously incorrect for the following reason: The English (or any other language you choose) has a finite number of words. The combinations resulting from fourteen selected words of a finite set is also finite. The natural numbers are infinite and therefore they cannot be assigned to all the combinations of 14 or fewer words , even if all these combinations had some meaning.

The following proof is by the method of reductio ad absurdum that works as follows: We want to prove that a proposition A is true. We assume initially that it is false and then with logical reasoning we try to reach a contradiction. Then the assumption we made that proposal A is false is not correct and therefore the sentence A must be true .
Try to discover why the proof is incorrect.

PROOF

1. Assume that there are natural numbers that cannot be described with fourteen words or less.
2. One of these numbers is the smallest. Let's call it N.
3. Then the number N can be defined as "the smallest natural number that cannot be described with fourteen words or less".
4. This proposal describes the number N with fourteen words or less and thus contradicts the hypothesis that the N is a number that cannot be described with fourteen words or less.
5. The initial assumption we made in Step 1 resulted with logical steps in contradiction so it must be false.
6. So all natural numbers can be determined with fourteen words or less!



*I have already failed to solve the challenge if that interests you.