For questions about well-orderings and well-ordered sets. Depending on the question, consider adding also some of the tags (elementary-set-theory), (set-theory), (order-theory), (ordinals).
A well-order is a linear order where every non-empty set has a minimal element. Equivalently, it is a partial ordering where every non-empty set has a minimum.
We say that a set $A$ is well-ordered if it comes with a well-order of $A$; and that it is well-orderable if there is a well-ordering of $A$. Assuming the axiom of choice (or equivalently, Zorn's lemma) every set is well-orderable. That is Zermelo's theorem.
Well-ordered sets are exactly the linear orders on which we can perform recursive definitions and inductive proofs. The simplest examples include all the finite linear orders, and the natural numbers.
On the other hand, the rational numbers, and the real numbers, are all examples of linear orders which are not well-ordered.
The study of well-orders goes hand in hand with the study of ordinals, which are order types of well-ordered sets, and the von Neumann ordinal assignment, which gives us a specific well-ordered set isomorphic to a given well-order.
