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
Best Answer
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.