Archimedean property

I've been studying the axiomatic definition of the real numbers, and there's one thing I'm not entirely sure about.

I think I've understood that the Archimedean axiom is added in order to discard ordered complete fields containing infinitesimals like the hyperreal numbers. Additionally, this property clearly cannot be derived solely from the axioms of ordered field and completeness, since $^*\mathbb{R}$ and $\mathbb{R}$ are two complete ordered fields, two models of the axioms, one of them Archimedean and the other non-Archimedean. Are these ideas correct?

Thanks.

Solution 1:

The answer to your question depends critically on what you mean by a "complete ordered field" $(F,<)$. Here are two rival definitions:

1) [added: sequentially] Cauchy complete: every Cauchy sequence in $F$ converges.

2) Dedekind complete: every nonempty subset $S \subset F$ which is bounded above has a least upper bound.

(There are in fact many other axioms equivalent to 2): that every bounded monotone sequence converges, that $F$ is connected in the order topology, the principle of ordered induction holds, and so forth.)

It turns out that there is a unique Dedekind complete ordered field up to (unique!) isomorphism, namely the real numbers $\mathbb{R}$. Famously $\mathbb{R}$ is also Cauchy complete -- or, if you like, Dedekind complete ordered fields satisfy the Bolzano-Weierstass theorem, which is enough to make Cauchy sequences converge -- so that Dedekind completeness implies Cauchy completeness.

The converse is true with an additional hypothesis: an Archimedean Cauchy-complete field is Dedekind complete. I show this in $\S 12.7$ of these notes using somewhat more sophisticated methods (namely Cauchy nets). For a more elementary proof, see e.g. Theorem 3.11 of this nice undergraduate thesis.

On the other hand, just as one can take the "Cauchy" completion of any metric space (or normed field) and get a complete metric space (or complete normed field), one can take the Cauchy completion of a non-Archimedean ordered field and get an ordered field which is Cauchy complete but not Dedekind complete. The easiest example of such a field is probably the rational function field $\mathbb{R}(t)$ with the unique ordering that makes $t$ positive and infinitely large.

For some reason these subtleties seem to be hard to find in standard analysis texts. I myself didn't learn about them until rather recently (so, several years after my PhD). I actually wrote up some of this material as supplemental notes for a sophomore-junior level course I am currently teaching on sequences and series...but I have not as yet been able to make myself inflict these notes on my students. I talked about ordered fields in several lectures and it seemed to be one level of abstraction beyond what they could even meaningfully grapple with (so it started to seem a bit pointless).

Solution 2:

There are non-archemedian completions of the rationals, called p-adic completions. The book Gouvêa, Fernando Q. (2000). p-adic Numbers : An Introduction (2nd ed.). Springer is an excellent introduction to these.