The Mathematics of Infinity: A Guide to Great Ideas, 2nd Edition

Chapter 6

Weil-Ordered Sets

The set of real numbers is linearly ordered. That is, given x, y ∈ then x ≤ y or y ≤ x. Such an ordering of a set X allows us to think of X as some part of a line. For instance, is linearly ordered since it is a subset of . It is the rational part of the real line. The integers, , form a linearly ordered set, but it has the very interesting property that to each there is a next integer or a successor to x, usually denoted by x⁺. One easily sees that x + 1 is the successor to x in . The existence of a successor for each element in is called the Well-Ordering Principle. We say that , is well-ordered. Because , is well-ordered, but is not well-ordered. In this chapter, ...