An abnormal man looking for an interesting number

After taking the picture below, from the building written “normal”, the abnormal guy of the photo — me — remembered an “interesting” paradox.

Suppose I list the natural numbers in order:

• 1
• 2
• 3
• 4

Now let’s say something interesting about each of these numbers:

• 1 is the first number of all, it is divisor of all the others
• 2 is the first and only prime number
• 3 is the first odd cousin
• 4 is the first perfect square

Let’s say the numbers with the an interesting property are called “interesting” numbers.
And the numbers that are not interesting are the “normal” numbers.

Using this definition, a list would look like this:

• 1 is an interesting number
• 2 is an interesting number
• 3 is an interesting number
• 4 is an interesting number

Now suppose the number x is the first “normal” number in the list.

• 1 is an interesting number
• 2 is an interesting number
• 3 is an interesting number
• 4 is an interesting number
• x is a normal number

But if x is the first “normal” number, it is an “interesting” number because it has an interesting property: to be the first “normal” number.

On the other hand, if we consider x an “interesting” number by having the property of being the first “normal” number, it is no longer a “normal” number and now it is an “interesting” number, this way losing the property of being the first “Normal” and then ceasing to be “interesting” …

What a mess! It’s not “interesting”?

To tell you something interesting, this problem is the “Paradox of Richard’s Numbers,” described by the mathematician Jules Richard in 1905.