4.4: The Well-Ordering Principle
The penultimate theorem of this chapter is known as the Well-Ordering Principle . As you shall see, this seemingly obvious theorem requires a bit of work to prove. It is worth noting that in some axiomatic systems, the Well-Ordering Principle is sometimes taken as an axiom. However, in our case, the result follows from complete induction. Before stating the Well-Ordering Principle, we need an additional definition.
Definition 4.35. Let \(A\subseteq \mathbb{R}\) and \(m\in A\) . Then \(m\) is called a maximum (or greatest element ) of \(A\) if for all \(a\in A\) , we have \(a\leq m\) . Similarly, \(m\) is called minimum (or least element ) of \(A\) if for all \(a\in A\) , we have \(m\leq a\) .
Not surprisingly, maximums and minimums are unique when they exist. It might be helpful to review Skeleton Proof 2.90 prior to attacking the next result.
Theorem 4.36. If \(A\subseteq \mathbb{R}\) such that the maximum (respectively, minimum) of \(A\) exists, then the maximum (respectively, minimum) of \(A\) is unique.
If the maximum of a set \(A\) exists, then it is denoted by \(\max(A)\) . Similarly, if the minimum of a set \(A\) exists, then it is denoted by \(min(A)\) .
Problem 4.37. Find the maximum and the minimum for each of the following sets when they exist.
- \(\{5,11,17,42,103\}\)
- \(\mathbb{N}\)
- \(\mathbb{Z}\)
- \((0,1]\)
- \((0,1]\cap \mathbb{Q}\)
- \((0,\infty)\)
- \(\{42\}\)
- \(\{\frac{1}{n}\mid n\in\mathbb{N}\}\)
- \(\{\frac{1}{n}\mid n\in\mathbb{N}\}\cup\{0\}\)
- \(\emptyset\)
To prove the Well-Ordering Principle, consider a proof by contradiction. Suppose \(S\) is a nonempty subset of \(\mathbb{N}\) that does not have a least element. Define the proposition \(P(n):=\) “ \(n\) is not an element of \(S\) " and then use complete induction to prove the result.
Theorem 4.38. Every nonempty subset of the natural numbers has a least element.
It turns out that the Well-Ordering Principle (Theorem 4.38) and the Axiom of Induction (Axiom 4.1) are equivalent. In other words, one can prove the Well-Ordering Principle from the Axiom of Induction, as we have done, but one can also prove the Axiom of Induction if the Well-Ordering Principle is assumed.
The final two theorems of this section can be thought of as generalized versions of the Well-Ordering Principle.
Theorem 4.39. If \(A\) is a nonempty subset of the integers and there exists \(\ell\in \mathbb{Z}\) such that \(\ell\leq a\) for all \(a\in A\) , then \(A\) contains a least element.
Theorem 4.40. If \(A\) is a nonempty subset of the integers and there exists \(u\in \mathbb{Z}\) such that \(a\leq u\) for all \(a\in A\) , then \(A\) contains a greatest element.
The element \(\ell\) in Theorem 4.39 is referred to as a lower bound for \(A\) while the element \(u\) in Theorem 4.40 is called an upper bound for \(A\) . We will study lower and upper bounds in more detail in Section 5.1.