All natural numbers are interesting

Theorem: All natural numbers are interesting.1

Proof by contradiction.

Assume the theorem is false. Then there must be at least one number \in \N that is boring.

Let \mathbb{B} \subset \N be the set of boring numbers.

By the assumption, we have:

\mathbb{B} \ne \phi\qquad{(1)}

Thus, because \mathbb{B} is non-empty \mathbb{B} \subset \N, the Well Ordering Principle tells us that:

\exists b \in \mathbb{B} \| b = \mathrm{inf}(\mathbb{B})\qquad{(2)}

Thus this b is the smallest boring natural number in existence; which makes b interesting. A contradiction! ∎

Note that the same doesn’t trivially hold for the set of real numbers since you’d first have to prove that the set of boring real numbers is either finite or bounded, and that the infimum is a member of the set. Neither does this hold for \Z since a non-finite subset in \Z doesn’t necessarily have a minimum.

  1. Not actually a theorem.↩︎