This is the well ordering principle as it is given in Thomas Koshy's number theory book -
Every non empty set of positive integers has a least element.
For example the set {17,23,5,18,13} has a least element 5.
By virtue of the well ordering principle, the set if positive integers in well-ordered. You may notice that the set of negative integers is not well ordered.
I didn't understand the statement - "You may notice that the set of negative integers is not well ordered".
A set of negative numbers also has a least number. Or is it that the well ordered principle is confined to only positive numbers and that's the reason the above statement holds good?
In the traditional ordering system we use for the integers. It is possible to set up an ordering of the integers where the entire set does have a least element. (But then we abandon the concept of positive and negative. I have never heard of an actual use for such an ordering.)
-Dan
Extreme care must be taken in putting this.
But if we accept the Axiom of Choice then one of its consequences/equivalences is the theorem that any set can be well ordered. But be careful, that theorem simply states: Given any set A there exists an order relation G such that G well orders A. Note that we have no way of knowing what G looks like in general. It is likely not .