Axioms for ℝ—in which we invent Arithmetic, Order our numbers and Complete our description of the reals

The previous chapter represents the culmination of what computing aficionados might call the ‘bottom-up’ approach to the real numbers. That is to say, we began with the natural numbers ℕ and constructed, successively, the integers ℤ, the rational numbers ℚ and finally the real numbers ℝ. Now we examine the ‘top-down’ approach and adopt a more algebraic stance. The real numbers will be defined by an abstract set of axioms from which we shall deduce (as theorems), at first elementary, and later more sophisticated, properties of ℝ. The advantage of the axiomatic approach is that it does not depend on any preconceived ideas of what real numbers are. However, some of our first ‘theorems’ will appear trivial to our trained minds—the point to bear in mind is that they are consequences of even more basic assumptions— namely, our axioms. To make the following exposition more palatable, we shall present the defining axioms in three measured helpings. We shall also enlist the aid of our friends Alice, Tweedledee and Tweedledum… so, to work!