Is it Possible to Apply the Well-Ordering Principle to a Subset of the Non-Negative Integers? Proof of the Division Algorithm
In many books on number theory they define the well ordering principle (WOP) as:
Every non-empty subset of positive integers has a least element.
Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible to apply the WOP to a subset of non-negative integers? Am I being too pedantic?
 A: Are you being too pedantic? You're raising a valid point, so no. Not everyone would care, but this is by no means extreme pedantry.
As to how they can apply the WOP, the positive integers and the non-negative integers (both under the standard ordering) are order isomorphic, add can be seen by adding or subtracting 1. Thus any ordering property of one applies to the other as well. Being well-ordered included.
A: Every non-empty subset of non-negative integers has a least element is also an acceptable way to phrase the well-ordering principle. (See a reference here for example.)
The statement applied to a subset of negative integers is however false. 
A: In David M. Burton's Elementary Number Theory text, the Well-Ordering Principle is given as follows: Every nonempty set $S$ of nonnegative integers contains a least element; that is, there is some integer $a$ in $S$ such that $a \leq b$ for all $b$'s belonging to $S$.
So, according to this definition, you may apply the Well-Ordering Principle to any nonempty set of non-negative integers.
Incidentally, I checked Rosen's number theory text and he uses the term 'positive' in lieu of 'non-negative.' This does not present a problem because if $S$ is a set of positive integers to which Rosen's statement of the Well-Ordering Principle applies, we may augment $S$ by inserting $0$ into it. Now, any non-empty subset of $S$ that contains $0$ must necessarily have $0$ as a least element, which is in $S$.     
