In this chapter we shall show how to construct a complete ordered field from a simple chain. This construction is important for two reasons. It proves that any contradiction inherent in the postulates for a complete ordered field, that is, the real number system, is latent in the postulates for a simple chain, which is a far less complicated structure whose consistency is almost guaranteed by our intuition. Perhaps more important to the novice in abstract mathematics is that it provides an excellent demonstration of set-theoretical techniques.