Characterizing the Real FieldThe real field consists of the set IR of real numbers, together with the operationsof addition and multiplication, the additive and multiplicative identities 0 and 1, andthe ordering relation ≤.We have come across sixteen properties of the real field. First, on page 2, there arethe nine field axioms, P1–P9. (In P7, it should be added that 0 6= 1 in order to obtaina correct list of axioms for fields.) Secondly, on page 2, there are the five orderingproperties, O1–O5. Thirdly, on page 4, we came to the Archimedian property. Andfinally, on page 48, we came to the “axiom of completeness,” stating that Cauchysequences converge.These sixteen properties completely characterize the real field, in a sense to bemade precise. That is, any complete Archimedian ordered field is exactly like (thetechnical term is “isomorphic to”) the real field.To state this more accurately, suppose we have some other set F , together withoperations + and × on F , additive and multiplicative identities 0Fand 1F, and anordering relation ≤F. We will say that F is a complete Archimedian ordered field ifthis structure satisfies all sixteen properties: the field axioms, the ordering axioms,the Archimedian property, and the axiom of completeness.Theorem. Assume that a set F , together with operations + and ×, elements0Fand 1F, and relation ≤F, is a complete Archimedian ordered field. Then thereis a one-to-one function h from IR onto F with the following properties (for all realnumbers x and y):(i) h “preserves addition”: h(x + y) = h(x) + h(y) (where the + sign on the rightside of the equation denotes addition in F , and the + sign on the left side denotesaddition in IR).(ii) h “preserves multiplication”: h(xy) = h(x) × h(y).(iii) h(0) = 0Fand h(1) = 1F.(iv) h is “order-preserving”: Whenever x ≤ y then h(x) ≤Fh(y)Moreover, there is a unique such function h.The function h is said to be an isomorphism from the real ordered field to the Fordered field, and we say that the two fields are isomorphic.A proof will not be given here, but we will see how the function h is built up.First of all, condition (iii) tells us to define h(0) = 0Fand h(1) = 1F. This getsus started.Secondly, condition (i) requires that we define, for example, h(3) = 1F+ 1F+ 1F,because 3 = 1 + 1 + 1. And similarly for any other natural number. So we now haveh(n) defined for each non-negative integer n.Thirdly, because 3 + (−3) = 0, we need to define h(−3) to be the additive inverseof h(3) in F . And similarly for the other negative integers.Fourthly, condition (ii) requires that we define, for example, h(5/3) to be theunique d in F for w hich d × h(3) = h(5). That is, h(5/3) = h(5) × h(3)−1, where onthe right side we take the multiplicative inverse in F . And similarly for other rationalnumbers. So we now have h(q) defined for each rational q.Finally, we come to the irrational real numbers . We know that any irrationalnumber r equals the supremum of the set of smaller rational numbers. The fieldF , being Archimedian and complete, has the leas t-upper-bound property, by §2.5.Condition (iv) requires that h is order-preserving, so we defineh(r) = sup{h(q) | q < r and q ∈ 1Q}where on the right side we take the supremum in F .This completes the construction of the isomorphism h. The proof that it is one-to-one, onto, and has properties (i)–(iv) is not given here.The point is that the sixteen properties give us a full picture of the structure ofthe real field. We know that non-Archimedian fields exist, which are very differentfrom the reals. We know that incomplete ordered fields exist, for example, the rationalfield. But there is only one complete Archimedian ordered field, up to isomorphism,and that is the real field.H. B.