This is a well-known joke, so many of you might already be familiar with it. Nonetheless, it’s quite amusing as a joke. 😆

The Assertion: “Everyone is Bald”

As indicated in the title, we will prove the assertion that “everyone is bald” using mathematical induction (often simply called induction).

Understanding Mathematical Induction

Let \( P(n) \) be a proposition about the natural number \( n \). Mathematical induction is a method that states:

1. The starting point, \( P(1) \) , is true.
2. For each natural number \( k \), if \( P(k) \) is true, then \( P(k+1) \) is also true.

If both 1. and 2. hold, then \( P(n) \) is true for any natural number \( n \).

Applying Induction to “Everyone is Bald”

Now, let’s apply this induction method to the statement above.

1. A person with only one hair is clearly bald.
2. If we consider a person with \( k \) hairs to be bald, then a person with \( k+1 \) hairs is also bald.

Therefore, by induction, it follows that everyone is bald.

Analyzing the Flaws in this Argument

However, this conclusion doesn’t quite match our actual sense of reality, so let’s address the flaw in this argument. Induction is a method used to prove “propositions” involving natural number variables.

A proposition is:

A statement that can be objectively judged as either true or false.

For example, “Mount Fuji is the tallest mountain in Japan” is a proposition, but “Mount Fuji is tall” is not.

The Subjectivity of Baldness

Since baldness is subjective, saying “a person with \( k \) hairs is bald” cannot be universally affirmed. In other words, “a person with \( k \) hairs is bald” is not actually a proposition, which means applying induction here was, in fact, impossible.