The uniqueness of the reals


1. Introduction

This web page is devoted to a very beautiful result that any two complete ordered fields are isomorphic. In fact, the result is even better than this and the isomorphism is canonical, in particular is unique and natural.

This is a sketch of the idea. Suppose F , G are two complete ordered fields and we are trying to construct an embedding of F in G . Then for reasons to do with the ordered field structure, the copy of the rationals in F and the copy of the rationals in G are the same so we must map F 's copy of onto G 's copy of in the obvious and only way possible.

Now for x F there is a monotonic nondecreasing sequence ( a n ) of rationals converging to x by one of the density theorems. Since we are forced to send a n to the copy of a n in G (as a n ) we must therefore send x to the limit of the copy of the sequence ( a n ) in G . Fortunately (by the completeness axiom in G ) this sequence does indeed have a limit so there is some copy of x in G to send x to. This construction will give a canonical embedding of F in G , and if F is also complete this embedding will turn out to be an isomorphism.