# 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, ...

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