4.1: Well-orderings
In this chapter we discuss the principle of mathematical induction. Be aware that the word induction has a different meaning in mathematics than in the rest of science. The principle of mathematical induction depends on the order structure of the natural numbers, and gives us a powerful technique for proving universal mathematical claims.
DEFINITION. Well-ordering Let \(X\) be a set, and \(\preceq\) a linear ordering on \(X\) . We say that \(X\) is well-ordered with respect to \(\preceq\) (or \(\preceq\) is a wellordering of \(X\) ) if every non-empty subset of \(X\) has a least element with respect to \(\preceq\) . That is, for any non-empty subset \(Y\) of \(X\) \[(\exists a \in Y)(\forall y \in Y) a \preceq y .\] In general, linear orderings need not be well-orderings. Well-ordering is a universal property — a set \(X\) with an ordering \(\preceq\) is well-ordered if every non-empty subset of \(X\) has a least element with respect to \(\preceq\) . If there is any non-empty subset which does not have a least element, then \(\preceq\) does not well-order \(X\) .
EXAMPLE 4.1. \(\mathbb{Z}\) is not well-ordered by \(\leq\) . The integers do not have a least element, which suffices to demonstrate that \(\mathbb{Z}\) is not well-ordered by \(\leq\) .
EXAMPLE 4.2. Let \(X=\{x \in \mathbb{R} \mid x \geq 2\}\) . Let \(\leq\) be the usual ordering on \(\mathbb{R} . X\) is linearly ordered by \(\leq\) , but \(X\) is not well-ordered by \(\leq\) . In this example, \(X\) has a least element, but any open interval contained in \(X\) will fail to have a least element. The key order properties of \(\mathbb{N}\) are that it is well-ordered and every element of \(\mathbb{N}\) , except 0 , is the successor of a natural number:
WELL-ORDERING PRINCIPLE FOR THE NATURAL NUMBERS: The set \(\mathbb{N}\) is well-ordered by \(\leq\) .
SUCCESSOR PROPERTY FOR THE NATURAL NUMBERS: If \(n \in \mathbb{N}\) and \(n \neq 0\) , then there is \(m \in \mathbb{N}\) such that \(n=m+1\) .
If one accepts an intuitive understanding of the natural numbers, these principles are more or less obvious. Indeed, let \(Y\) be any nonempty subset of \(\mathbb{N}\) . Since it is non-empty, there is some \(m\) in \(Y\) . Now, consider each of the finitely many numbers \(0,1,2, \ldots, m\) in turn. If \(0 \in Y\) , then 0 is the least element. If 0 is not in \(Y\) , proceed to 1 . If this is in \(Y\) , it must be the least element; otherwise proceed to 2 . Continue in this way, and you will find some number less than or equal to \(m\) that is the least element of \(Y\) .
This argument, though convincing, does rely on the fact that we have an idea of what \(\mathbb{N}\) "is". If we wish to define \(\mathbb{N}\) in terms of set operations, as we do in Chapter 8, we essentially have to include the well-ordering principle for the natural numbers as an axiom.