What's the difference between the Well-ordering principle and Least Upper Bound property?

Question

Well-Ordering Principle : Every non-empty set of positive integers contains a least element

Least Upper Bound Property: Every nonempty bounded subset of the real numbers has a least upper bound.

Is the Well-Ordering derived from the LUB property?

Answer 1

I would not derive a basic property of the natural numbers, like the well-ordering principle, from a more complicated property of the more complicated system of real numbers, like the least upper bound principle. I believe that most if not all systematic presentations of the relevant foundations agree with me and prove well-ordering of the natural numbers before doing anything with the real numbers.

To also answer your title question: The difference is that (1) they are about different systems --- the natural numbers in one case and the real numbers in the other --- and (2) they assert very different things about these systems --- existence of a smallest element of a set in the one case and existence of an element "near" the top of a set in the other.

Perhaps a better question would be what are the similarities between these two principles. I can think of just two: First, any well-ordering has the least-upper-bound property, even without the "nonempty" hypothesis (but note that this does not help in proving the least-upper-bound property for the real numbers, as these are not well-ordered). Second, both properties involve a quantifier over arbitrary subsets of the system, so they cannot be expressed fully in first-order logic but need second-order logic.