Interesting Number Paradox

Paradox

Some numbers have interesting properties.

For example: $F_{12} = 144$ is a square Fibonacci number which also happens to be the square of its index.

Some numbers, on the other hand, have no such interesting properties.

Suppose all the positive integers were partitioned into $2$ sets: the interesting numbers $I$ and the uninteresting numbers $U$.

Let $s \in U$ be that smallest element.

Then $s$ has an interesting property: it is the smallest uninteresting number.

Thus $s$ is not an element of $U$, but of $I$.

It follows that the smallest element of $U$ is not $s$ but $s'$ where $s' > s$.

But then $s'$ is now the smallest uninteresting number, and therefore interesting.

And so on.

It follows that there are no uninteresting numbers.

This is an antinomy.

The integers in $U$ are defined as uninteresting.

Once $U$ has been defined, redefining $s$ as interesting cannot be done.

Saying that $s$ is interesting because it is the smallest element of $U$ is to say that $U$ is improperly defined.

Ultimately, the point is that $U$ cannot be well-defined.

If $s$ is moved from $U$ to $I$, that which makes it interesting (it being the smallest element of $U$) no longer holds.

$s$ is no longer interesting and so no longer belongs to $I$.

Thus $s$ cannot belong either to $U$ or to $I$ and hence neither $U$ nor $I$ can be defined.

$\blacksquare$