Well-ordering Principle

Every set can be Well-ordered.

Proof

Let
be Partial Ordered by extension
iff , , is an initial segment of
Note , e.g.
Let be a nonempty Chain in , i.e. a nested set of Well-ordered sets.
Then is an Upper Bound for
By Zorn’s Lemma there is a Maximal element of .
If then for … is a strictly greater than