Proof of the Division Algorithm

[matqkks]

TL;DR

Application of well ordeing principle

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?

 

[andrewkirk]

Yes, you can apply it to the non-negative integers, by simply observing that if the subset contains zero then zero is the least element, otherwise the subset is also a subset of the positive integers and we can apply the principle that holds for them.

 

[member 587159]

Summary: Application of well ordeing principle

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?

Yes, well ordering principle applies to any subset of $\mathbb{Z}$ that is bounded below.

 

[matqkks]

Thanks it is so obvious as you have suggested.