Full circle

We began

We began this chapter with an example of an ordered field for which the intermediate value theorem failed. A simple extension of that example shows that just as the fundamental axiom implies the intermediate value theorem, so the intermediate value theorem implies the fundamental axiom.

Are the real numbers unique?

Our statement of Theorem 1.53 raises the question as to whether there may be more than one ordered field satisfying the fundamental axiom. The answer, which is as good as we can hope for, is that all ordered fields satisfying the fundamental axiom are isomorphic. The formal statement is given in the following theorem.