Every nonempty set of nonnegative integers has a smallest element.
This statement is known as The Well Ordering Principle. Do you believe it? Seems sort of obvious, right? But notice how tight it is: it requires a nonempty set—it’s false for the empty set which has no smallest element because it has no elements at all. And it requires a set of nonnegative integers—it’s false for the set of negative integers and also false for some sets of nonnegative rationals—for example, the set of positive rationals. So, the Well Ordering Principle captures something special about the nonnegative integers.
While the Well Ordering Principle may seem obvious, it’s hard to see offhand why it is useful. But in fact, it provides one of the most important proof rules in discrete mathematics. In this chapter, we’ll illustrate the power of this proof method with a few simple examples.