A Discussion of the Fallacy

It is very difficult to pin down this fallacy. It concerns the informal nature of the English language, and the paradoxes that can arise from self referencing statements.  Any sentence that make claims about itself runs the risk of collapsing the proof system itself resulting in apparent inconsistencies. The classic example is the liar's paradox "this sentence is false".

The problem in this case is the following phrase (let's call it S):

the smallest natural number that cannot be unambiguously described in fourteen words or less.
S refers to itself (because it makes claims about descriptions of numbers, and S is such a description). Moreover, it does so in a logically inconsistent fashion: if you try to apply the description S to a number, then it ends up stating that S does not apply to that number.

This means that S cannot be considered as a self-consistent description of any natural number. This, however, does not mean that the number n (in the proof) does not exist! There is such a number n, and n is the smallest natural number that cannot be unambiguously described in fourteen words or less; it's just that the phrase "the smallest natural number that cannot be unambiguously described in fourteen words or less" is not a description of it (in the sense that is being used in the proof), because it is a phrase that cannot be self-consistently asserted about any number.

Therefore, step 4 of the proof (which mistakes the self-inconsistent nature of S with a mathematical contradiction arising from the existence of n) is at fault.

This is also related to Russell's Paradox in set theory: there is no such thing as the "set of all sets" (if there were, you could look at "the set of all sets that do not contain themselves". Let S be this set. Does S contain itself, or not? Either way leads to a contradiction).

Finally, although this particular proof is fallacious, it illustrates a common proof technique which, when used correctly, is very powerful: the well-ordering principle.

If you want to show that something is true for all natural numbers n, one way to do it (which is mathematicall equivalent to a proof by induction but is sometimes more convenient than it) is to reason as follows:

Suppose it's not the case that the statement in question is true for all n. Then there is a smallest n for which it fails. But this leads to a contradiction because . . . . Therefore, it must indeed be the case that the statement in question is true for all n.

Proofs that follow this pattern are using the well-ordering principle (which says that any non-empty set of natural numbers must have a smallest element), and this is a very common and powerful pattern of proof. When used correctly, that is.