# Well-Ordering Principle

When we seek to develop the integers axiomatically (from a short list of basic assumptions), we usually include the Well-Ordering Principle as one of these assumptions:

- Well-Ordering Principle
- Every nonempty set S of positive integers contains a least
element; that is, there is some element
*a*of S such that*a*≤*b*for all elements*b*of S.

Notice that the positive real numbers do not have this
property. For example, there is no smallest positive real
number *r*, because *r*/2 is a smaller positive
real number! The negative integers also lack this
property because if *r* is a negative integer, then
*r*-1 is a smaller negative integer.

This simple principle of positive integers has many consequences. Let us demonstrate one by proving the following:

Theorem: Every integerngreater than one can be written as a product of primes.

This factorization is also unique (up to the order of the factors), see the Fundamental Theorem of Arithmetic.Proof: Eithernis prime (in which case we are done because it is the product of the one primen), or it has a positive divisor other than one and itself. Letp_{1}be the least of these divisors. Notice thatp_{1}must be prime, otherwise there is an integerkwith 1 <k<p_{1}, andkdividesp_{1}, sokdividesn, which contradicts the choice ofp_{1}! Son=p_{1}n_{1}wherep_{1}is prime andn>n_{1}.Now we repeat this argument with

n_{1}to find out that it is either prime (and we are done withn=p_{1}n_{1}), orn_{1}=p_{2}n_{2}, wherep_{2}is prime andn_{1}>n_{2}.Now again repeat the argument with

n_{2}, ... This process can not continue indefinitely because by the Well-Ordering Principle, the set of positive integershas a least element, sayn>n_{1}>n_{2}>n_{3}. . .p. Then_{k}n=p_{1}p_{2}^{.}...^{.}p. ∎_{k}