# Relationship between completeness and well ordering (meta).

Here is the definition for completeness of the reals (there are many equivalent formulations but I am interested in this one);

Completeness: Every non-empty subset of the reals which is bounded above has a least upper bound.

This can be translated to the following: Suppose $\displaystyle A$ is a non-empty set of real numbers bounded above. Then the set $\displaystyle B = \{ b \in \mathbb{R} : (\text{ for all } a \in A)(b \ge a)\}$ has a minimum element.

In other words, given a non-empty (bounded above) subest of real numbers, we can produce a natural set which has a minimum element.

Well ordering: Every non-empty subset of the natural numbers has a least element.

In other words, given a non-empty subest of natural numbers, we can produce a natural set which has a minimum element.

Question: Is this a coincidence, or is there some deeper structure or theory which can encapsulate this kind of behaviour, for the formulations are very close (almost) identical, i.e. beginning with an arbitrary subset which is non-empty and producing a minimum element?

You argument basically shows that a well-ordered set is always complete (in this definition): if a subset $A$ (of a well-ordered set $X$) has an upper bound, its set of upper bounds is non-empty so has a minimum by the well-orderedness. So $X$ is complete. So there is a connection between these notions, as you noted.