Paradoxical Proof?

[Apjjm]

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..

The fallacy

Spoiler below!

The problem is that the number referred to by the sequence "the smallest natural number that cannot be described with fourteen words or less" (this sequence will be referred to as Y from now on) actually refers to more than one number. Lets assume for the moment we use every other sequence of words to represent a number, and the smallest remaining number we have not represented is X. Then sequence Y must represent X, as it cannot be defined by our other sequences. However, as Y now can represent X, X is no longer unrepresentable in 14 or fewer words, and as such X+1 (or some number >x) is the smallest number and Y must refer to it. This means that interpreting Y in this manner makes Y inconsistent, and hence this interpretation of Y is incorrect.

tldr version: The phrase is self referential, hence inconsistent. The confusing behavior comes from the fact that the phrase represents multiple numbers when it should only represent one.
See also: Russells paradox

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.

Edit hint: So, here by stripping all the meaning from the words and just directly mapping them to integers we do not have any surprising results. This gives you a clue: using the english interpretation of the statements allows you to seemingly refer to a set of numbers larger than it should - you must therefore be somehow referring to more than one number with certain statements, violating the implicit constraint that the mapping must be bijective. The question then becomes which step of the process is worded such that this inconsistency can slip through, and how does this wording cause the problem.