*( Content warning: maths.
This was written as an exercise in approaching the underlying ideas with rigour and understanding. If it's of any help to other people, this is a happy side effect.)*

In discrete mathematics, the *Well-Ordering Principle* for natural numbers states that any non-empty set of natural
numbers contains a least element of itself

.

Slightly more formally: \[\forall S \subseteq \mathbb{N} : \; S = \emptyset \lor \exists x \in S (\forall y \in S (x \leq y))\]

**Example:** Let \(S = \{12,3,189,40,25\}\). Then the least element of \(S\) is \(3\).

**Example:** Let \(S = \{\}\). The Well-Ordering Principle doesn't apply because \(S\) is empty.

**Example:** Let \(S = \mathbb{N}\). Then the least element of \(S\) is \(0\).

**Example:** Let \(S = \{n \in \mathbb{N} \;|\; n\textrm{ odd}\} = \{1,3,5,\ldots\}\). Then the least element of \(S\) is \(1\).

The interesting thing here is that this applies to *every* subset of \(\mathbb{N}\), of which there are uncountably many. Some of them we can't compute membership for, e.g. the image of the busy beaver function. Most of them we can't even give a finite description for.