Another Proof of the Existence a Dedekind Complete Totally Ordered Field

We describe the Dedekind cuts explicitly in terms of non-standard rational num­ bers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the real numbers. We believe that our construction is simpler and shorter than the classical Dedekind construction and Cantor construction of such fields assuming some basic familiarity with non-standard analysis. 1 Preliminaries: Ordered Fields and Infinitesimals We recall the main definitions and properties of totally ordered rings and fields. We also recall the basic properties of infinitesimal, finite and infinitely large elements of such fields. For more details and for the missing proofs, we refer the reader to (Lang [4], Chapter XI), (van der Waerden [8], Chapter 11) and Ribenboim [6]. 1.1 Definition (Orderable Ring). Let K be a ring (field). Then: 1. K is called orderable if there exists a non-empty set K+ ⊂ K such that: (a) 0 ∈ K+; (b) K+ is closed under the addition and multiplication in K; (c) For every x ∈ K either x = 0 or x ∈ K+ or − x ∈ K+. �n 2 2. A ring (field) K is formally real if, for every n ∈ N, the equation k=0 xk = 0 in K has only the trivial solution x1 = · · · = xn = 0.