Examples of a complete ordered field

Question

We know that every complete ordered field is isomorphic to $\mathbb R$, but are there examples of complete ordered fields different, not isomorphically different of course, from $\mathbb R$?

Answer 1

If $\mathbb R$ denotes a complete ordered field, then the set of pairs $(x,0)$, where $x\in\mathbb R$, can also be regarded as a complete ordered field via the operations

• $(x,0)+(y,0)=(x+y,0)$
• $(x,0)\cdot(y,0)=(xy,0)$
• $(x,0)\le (y,0)$ iff $x\le y$

A less trivial example would be: as additive groups, $\mathbb R$ and $\mathbb C$ are isomorphic. Let $\varphi:\mathbb R\to\mathbb C$ be a group isomorphism. Use this isomorphism to define $\cdot$ and $\le$ on $\mathbb C$ in the following way: $\varphi(a+b)=\varphi(a)+\varphi(b)$, and $\varphi(x)\le\varphi(y)\iff x\le y$. Since $\varphi$ is bijective, these operations are well-defined. Then, $\varphi$ is an isomorphism of ordered fields.

More generally, any set with the same cardinality as $\mathbb R$ can be endowed with operations that turn it into a complete order field, via transport of structure.

Answer 2

Any Dedekind-complete ordered field can be defined to be the reals, $R$, although it is sometimes useful to have some other relationships between the members of $R. $ Examples: Assume that we have "the" field $Q$ of rationals, we can define $R$ as the set of equivalence classes of Cauchy sequences in $Q$, or as the union of $Q$ with the set of its proper Dedekind cuts, or by the usual set of decimal representations....