Proof of the division algorithm

[matqkks]

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?

 

[Dr.Peterson]

Since the principle as you state it requires positive integers, it can't be applied directly.

But given a set [MATH]A[/MATH] of non-negative integers, you could form the set [MATH]B = \{a+1 : a\in A\}[/MATH], and apply the principle to that.

 

[matqkks]

Thanks.