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:
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 integer n greater than one can be written as a product of primes.
Proof: Either n is prime (in which case we are done because it is the product of the one prime n), or it has a positive divisor other than one and itself. Let p1 be the least of these divisors. Notice that p1 must be prime, otherwise there is an integer k with 1 < k < p1, and k divides p1, so k divides n, which contradicts the choice of p1! So n = p1n1 where p1 is prime and n > n1.This factorization is also unique (up to the order of the factors), see the Fundamental Theorem of Arithmetic.