## 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.