TITLE: There are no bearded men in the world - What goes wrong in this proof?
QUESTION [19 upvotes]: Several years ago in a textbook I read this example as a faulty use of proof by induction. I never really realized why it fails. Here it goes:

Theorem. There are no bearded men in the world.

Proof by induction

Base case: Suppose a person has n=1 facial hair. That's not enough to
  be called a beard.
Induction step: Assume as induction hypothesis that the statement
  holds true for n = k hair, meaning the person has n = k facial hair
  that are not enough to constitute a beard. Adding one hair to the set
  would not matter and the statement would still hold true.
Therefore no bearded man exists in the world.

What's the flaw here?

REPLY [2 votes]: There are three ways to disagree with an argument:
1 - Identify an ambiguous or incorrectly-used term
2 - Disagree with the premise
3 - Show that the conclusion does not actually follow from the premises
In this case, the simple reason to argue it's wrong is that disagree with the premise you stated: "Assume as induction hypothesis that the statement holds true for n = k hair"
The slightly more complex reason is the implied assumption that the number of individual hairs is what determines the presence or absence of a beard.