Step 1. Suppose there is at least one counting number that cannot be unambiguously described in 14 words or less.
Step 2. Let N be the smallest such number.
Step 3. N is then "the smallest counting number that cannot be unambiguously described in 14 words or less".
Step 4. The words above in quotes are a 14-word description of N that define it unambiguously.
Step 5. Therefore we have contradicted the statement in Step 2. Hence our original assumption, ie Step 1, must be false.
Let us list all the possible combinations of 14 words. One of the combinations in our list is "the smallest counting number that cannot be unambiguously described in 14 words or less". Let's refer to it as "D". What D is saying is that our number N cannot be unambiguously described by any of the statements in the list. Since D itself is in the list, this means that D is also referring to itself. In effect D says that N is the smallest number not described by any of the combinations in the list, including that N is not described by D. Yet D purports to unambiguously describe N.
So D is saying two things: 1) N is the number having the property explicitly stated by D (ie "the smallest counting number that cannot be unambiguously described in 14 words or less"), and that 2) N is a number not specified by D. In other words the specification given by D is self-contradictory. It is as though D is simultaneously saying, "N is zero" and "N is non-zero".
D is a variant of the self-referential paradox, eg "This statement is false" (also known as the "liar's paradox"), where 'this' is taken to refer to the statement itself. It has the form of a sensible statement but not the content. Essentially it says, "I deny what I assert" but does not actually assert anything. The moral of the story is that self-reference in logic is almost as dangerous as division by zero in arithmetic. In many cases it simply does not make sense. See Liar's Paradox for a fuller discussion.
So which step is at fault in the original 'proof'? Step 4 is at fault because it appears to be a sensible description, whereas it is the equivalent of "white is black". In other words, D, the description given in Step 3, does not unambiguously describe a number simply because it is self-contradictory.
Finally, what can we conclude about N - is there such an animal? Yes, N exists. To discover it we would have to find all the numbers validly described by our list of 14-word descriptions. Note that the list of descriptions should exclude D, as well as all other nonsensical or not number-specifying combinations. N is then indeed "the smallest counting number that cannot be unambiguously described in 14 words or less". The only point to remember is that the words in quotes are not in the list of descriptions.
(My guess is that N is a fifteen-digit number.)
The possibility that Cantor's diagonal procedure is a paradox in its own right is not usually entertained, although a direct application of it does yield an acknowledged paradox: Richard's Paradox. Consider for a start all finite sequences of the twenty six letters of the English alphabet, the ten digits, a comma, a full stop, a dash and a blank space. Now some of these expressions are English phrases, and some of those phrases will unambiguously define real numbers. Order the latter expressions lexicographically. We then have a way of identifying the n-th member of this collection.
The following phrase, as Richard pointed out, would seem to define a real number which is not defined in the collection:
'The real number whose whole part is zero, and whose n-th decimal place is p plus 1 if the n-th decimal of the real number defined by the n-th member of our collection is p, where 9 goes to 0.' But this expression is a finite sequence of the previously described kind.
This is a more sophisticated form of the 14-word paradox. The cause is the same - self-reference. You might want to think about the solution before reading on.
Answer:
The quoted expression is in the original list, hence when the diagonalisation procedure (adding 1 to the n th digit) is performed on the number defined by the quote we get a new number. In other words, the number defined by the quoted expression is different from itself. This is another variant of "I deny what I assert". QED