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.
QED
In case you think that's OK, remember that there is only a finite number of words in English and that therefore all 14-word combinations of these words must also be finite. Yet the counting numbers are infinitely numerous.