Paradoxical Proof?

[BAndrew]

There are a few ways to solve this problem. First an assumption which was unstated (this doesn't actually help pin down the fallacy, it just removes a little weasel out):
A word is a finite sequence of characters, drawn from a finite alaphabet..


Refutation without pointing out the fallacy

Spoiler below!

If we start off with a finite set of finite words, where each word is comprised of a finite alphabet. Given this, we can construct a finite dictionary D, of length #D (a finite integer) that holds every word.

It follows, that by replacing each word with it's index into D we can obtain a sequence of integers, up to 14 integers in length, where each integer is smaller than #D and greater than 0.

Therefore, the number of possible permutations of words is #D^14 + #D^13 + ... + #D^2 + #D which is finite, as each term in that expression is finite. This is important, as this number represents the maximum total number of integers we can represent using a 1-to-1 mapping (we actually might be able to represent less if we include restrictions on sentence structure!). This number is finite, hence there is not enough informational content to represent the countable infinity of the natural numbers without ambiguity.

I only saw your second part (Refutation without pointing out the fallacy) which was in a way stated in my post (or at least that's what I meant):

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.

Thanks though for the response. I'll wait to hear more opinions (if any) and then I'll read your answer.

---

To say that the statement "a number N cannot be described with 14 words or less" is a description of that number N is self-contradictory.(1) It's like


the sentence below is true
the above sentence is false
(2)

I didn't understand why (1) is similar to (2)