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?
Best Answer
The (lack of a) definition of what constitutes a beard is the flaw.