This chapter is devoted to various aspects of the structure of ℝ, the field of real numbers. Since we do not intend to give a detailed account of a construction of the real numbers from the very beginning, we need to clarify the basis of our subsequent arguments. What we shall assume about the real numbers is that they form an ordered field whose ordering is complete, in the sense that every non-empty bounded set of real numbers has a least upper bound. All these notions will be explained in due course, but presumably they are familiar to most readers. It is well known and will be proved in Section 11 that the properties just mentioned characterize the field of real numbers.
Historically, a satisfactory theory of the real numbers was obtained only at the end of the nineteenth century by work of Weierstraβ, Cantor and Dedekind (see Flegg 1983, Ehrlich 1994 and López Pellicer 1994). Starting from the rational numbers, they used different approaches, namely, Cauchy sequences on the one hand and Dedekind cuts on the other.
In Sections 42 and 44, we shall actually show how to obtain the real numbers from the rational numbers via completion. Another construction in the context of non-standard real numbers will be given in Section 23. We mention also the approach of Conway 1976, whose ‘surreal numbers’ go beyond non-standard numbers. These ideas were carried further by Gonshor 1986, Alling 1987; see also Ehrlich 1994, 2001 and Dales-Woodin 1996.