## 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.

This link (https://en.wikipedia.org/wiki/Richard%27s_paradox) tells more details about Richard’s paradox, but in a less interesting way than here.

Another similar paradox is the “Liar Paradox”. A man who only tells lies says “I’m lying.” But as he only lies, he will be telling the truth in this statement. But if he speaks the truth, he is not the one who only tells lies.

These paradoxes “bugs” not only the minds of ordinary human beings, but also the minds of the greatest mathematicians in history.

The Austrian mathematician Kurt Godel demolished the foundations of all mathematics in 1931, with its Incompleteness Theorems, by proving that mathematics can not at the same time be Complete and Consistent. That is, mathematics has limits. Godel found a “bug” in the foundations of mathematics — it can not at the same time get rid of these bizarre paradoxes and answer True or False to all its propositions. Godel used a sophisticated version of Richard Paradox to prove it.

It’s a long story, which involves mind giants like David Hilbert and Bertrand Russell, and it’s for another day.

By the way, I think the author of the building light wrote “normal” in an “interesting” way only for the building not to be “normal” anymore, and thus to confuse our head …