Every Number Is Special In Its Own Special Way

I got a question recently about where in the .NET framework the "special numbers" were defined. The questioner was actually asking about the Double.NaN, Double.PositiveInfinity, etc, special values for floating point numbers. Of course there are other "special numbers" defined by the framework, such as Math.PI.

The question was easily answered but it got me thinking, which, as we know, is usually trouble.

Clearly zero is a very special number, being the first natural number.

One is pretty special too, being the multiplicative identity.

Two is the only even prime.

Three is the lowest odd prime...

Clearly lots of numbers are special. This led me to propose the following theorem:

Theorem: Every natural number (0, 1, 2, ...) is a special number.


Let’s posit that the set of nonspecial natural numbers is nonempty, and deduce a contradiction.

If there exists a nonspecial natural number then there must be a lowest nonspecial natural number.

What an unusual property! The lowest nonspecial natural number!

Whatever number has that unusual property must be kinda... special.

Therefore the lowest nonspecial natural number is special.

Therefore, if the set of nonspecial natural numbers is nonempty then it contains a special number.

That is clearly nonsensical, therefore the set of nonspecial natural numbers is empty.

Therefore all natural numbers are special, QED.

And yet I can't find anything special about 7920687935872092847630945767548023. But it must be special somehow!

Extending the proof show that all real numbers are special is left as an exercise for the reader.