authored by Premmi and Beguène
Well-Ordering Principle
Every nonempty set S of nonnegative integers contains a least element, i.e., there is some integer a in S such that a \leq b for all b\text{'s} belonging to S.
Since we cannot prove the well-ordering principle by using properties that the nonnegative integers satisfy under the operations of addition and multiplication, we will consider the well-ordering principle as an axiom. Surprisingly enough, the well-ordering principle is logically equivalent to the principle of mathematical induction. This means that we if consider one as an axiom, you can use that to prove the other. We will prove this logical equivalence at a later time.
We will use the well-ordering principle in number theory to prove many important assertions with regard to integers. For example, the Division Algorithm uses the well-ordering principle to prove the existence of a nonnegative integer remainder greater than or equal to zero when an integer is divided by another positive integer.
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