Well-ordering principle

In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element.<ref>Template:Cite book</ref> In other words, the set of nonnegative integers is well-ordered by its "natural" or "magnitude" order in which <math>x</math> precedes <math>y</math> if and only if <math>y</math> is either <math>x</math> or the sum of <math>x</math> and some nonnegative integer (other orderings include the ordering <math>2, 4, 6, ...</math>; and <math>1, 3, 5, ...</math>).

The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem", according to which every set can be well-ordered. On other occasions it is understood to be the proposition that the set of integers <math>\{\ldots, -2, -1, 0, 1, 2, 3, \ldots \}</math> contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element.

Properties

Depending on the framework in which the natural numbers are introduced, this (second-order) property of the set of natural numbers is either an axiom or a provable theorem. For example:

• In Peano arithmetic, second-order arithmetic and related systems, and indeed in most (not necessarily formal) mathematical treatments of the well-ordering principle, the principle is derived from the principle of mathematical induction, which is itself taken as basic.
• Considering the natural numbers as a subset of the real numbers, and assuming that we know already that the real numbers are complete (again, either as an axiom or a theorem about the real number system), i.e., every bounded (from below) set has an infimum, then also every set <math>A</math> of natural numbers has an infimum, say <math>a^*</math>. We can now find an integer <math>n^*</math> such that <math>a^*</math> lies in the half-open interval <math>(n^*-1,n^*]</math>, and can then show that we must have <math>a^* = n^*</math>, and <math>n^*</math> in <math>A</math>.
• In axiomatic set theory, the natural numbers are defined as the smallest inductive set (i.e., set containing 0 and closed under the successor operation). One can (even without invoking the regularity axiom) show that the set of all natural numbers <math>n</math> such that "<math>\{0,\ldots,n\}</math> is well-ordered" is inductive, and must therefore contain all natural numbers; from this property one can conclude that the set of all natural numbers is also well-ordered.

In the second sense, this phrase is used when that proposition is relied on for the purpose of justifying proofs that take the following form: to prove that every natural number belongs to a specified set <math>S</math>, assume the contrary, which implies that the set of counterexamples is non-empty and thus contains a smallest counterexample. Then show that for any counterexample there is a still smaller counterexample, producing a contradiction. This mode of argument is the contrapositive of proof by complete induction. It is known light-heartedly as the "minimal criminal" method<ref>Template:Cite book</ref> and is similar in its nature to Fermat's method of "infinite descent".

Garrett Birkhoff and Saunders Mac Lane wrote in A Survey of Modern Algebra that this property, like the least upper bound axiom for real numbers, is non-algebraic; i.e., it cannot be deduced from the algebraic properties of the integers (which form an ordered integral domain).

Example applications

The well-ordering principle can be used in the following proofs.

Prime factorization

Theorem: Every integer greater than one can be factored as a product of primes. This theorem constitutes part of the Prime Factorization Theorem.

Proof (by well-ordering principle). Let <math>C</math> be the set of all integers greater than one that cannot be factored as a product of primes. We show that <math>C</math> is empty.

Assume for the sake of contradiction that <math>C</math> is not empty. Then, by the well-ordering principle, there is a least element <math>n \in C</math>; <math>n</math> cannot be prime since a prime number itself is considered a length-one product of primes. By the definition of non-prime numbers, <math>n</math> has factors <math>a, b</math>, where <math>a, b</math> are integers greater than one and less than <math>n</math>. Since <math>a, b < n</math>, they are not in <math>C</math> as <math>n</math> is the smallest element of <math>C</math>. So, <math>a, b</math> can be factored as products of primes, where <math>a = p_1p_2...p_k</math> and <math>b = q_1q_2...q_l</math>, meaning that <math>n = p_1p_2...p_k \cdot q_1q_2...q_l</math>, a product of primes. This contradicts the assumption that <math>n \in C</math>, so the assumption that <math>C</math> is nonempty must be false.<ref name='mit' >Template:Cite book</ref>

Integer summation

Theorem: <math>1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2}</math> for all positive integers <math>n</math>.

Proof. Suppose for the sake of contradiction that the above theorem is false. Then, there exists a non-empty set of positive integers <math>C = \{n \in \mathbb N \mid 1 + 2 + 3 + ... + n \neq \frac{n(n + 1)}{2}\}</math>. By the well-ordering principle, <math>C</math> has a minimum element <math>c</math> such that when <math>n = c</math>, the equation is false, but true for all positive integers less than <math>c</math>. The equation is true for <math>n = 1</math>, so <math>c > 1</math>; <math>c - 1</math> is a positive integer less than <math>c</math>, so the equation holds for <math>c - 1</math> as it is not in <math>C</math>. Therefore,

<math display=middle>\begin{align} 1 + 2 + 3 + ... + (c - 1) &= \frac{(c - 1)c}{2} \\ 1 + 2 + 3 + ... + (c - 1) + c &= \frac{(c - 1)c}{2} + c\\ &= \frac{c^2 - c}{2} + \frac{2c}{2}\\ &= \frac{c^2 + c}{2}\\ &= \frac{c(c + 1)}{2} \end{align}</math>,

which shows that the equation holds for <math>c</math>, a contradiction. So, the equation must hold for all positive integers.<ref name='mit' />