Well-ordering principle and negative integers

Question

The Wikipedia article on the Well Ordering Principle defines it [1] as:

"The well-ordering principle states that every non-empty set of positive integers contains a least element."

And it defines "least element" as "the least element of S is a lower bound of S that is contained within this subset. It is necessarily unique."

I understand it means that for every subset of R there is always a minimum element contained in the subset. Eg: {1, 2, 3} contains a least element, 1. {4, 5, 6} contains a least element 4, etc. But with this interpretation I fail to see is how this doesn't apply to the negative integers: the set {-1, -2, -3} contains a least element -3, {-4, -5, -6} contains a least element -6, etc.

If I interpret "contains" as "the set has a unique least element associated with it not necessarily present in the subset" then I understand that "least element" refers to the lowest upper bound of 1. And then I can see why the negative numbers don't have a least element (infinity). But it sounds to me too big of a leap of faith to interpret the wording this way.

I guess my question is: "What does "contains" exactly mean in the definition of the WOP?" Or "Why isn't it true that every non-empty set of negative integers contains a least element?"

Thanks

[1] http://en.wikipedia.org/wiki/Well-ordering_principle

Answer 1

Consider the set $\{n\in\Bbb Z : n<0\}$.

The sets in question do not have to be finite subsets. Every finite set of a total ordered set has a minimal element, the question is only interesting if the subsets are arbitrary.

Now, there are orderings under which $\Bbb Z$ is well-ordered, (consider any bijection wtih $\Bbb N$, for example) but your question is on the WOP as applied to $\Bbb N$ extended to $\Bbb Z$, so that's what I'm resticting my attention to.

Answer 2

One way to think about this is that the positives and negatives are a mirror. The "least element of the positives," can be thought of as the "greatest element of the negatives." You run into issues with infinity as you said, but you won't when you get close to 0. So the greatest negative is the closest to positive, which is the same as the least positive, if we ignore sign. I hope this helps at least somewhat