Chapter: Order

The order relations !\( \le \)", respectively "\( \ge \)" define a total order on the set of natural numbers \(\mathbb N\), i.e. any two natural numbers can be compared with each other. This is an important concept, since it enables one of the most important methods of mathematical proofs, called the principle of complete induction. Also we use this concept intuitively on a daily basis when comparing the number of things, paying or exchanging money, etc.

  1. Definition: Set-theoretic Definition of Order Relation for Natural Numbers
  2. Definition: Order Relation for Natural Numbers
  3. Proposition: Well-Ordering Principle of Natural Numbers
  4. Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers

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