Should one prove all Dedekind-complete ordered fields are isomorphic before using the real numbers in any formal matter?

Question

As mentioned in an early post I spent about nine hours cumulatively trying to construct the reals axiomatically from the bottom up using naive set theory. Though eventually I realized I made an error and tried fixing it, though this just made things more complicated and now I've spent several days doing this. Though now on the subject it got me thinking if I know the construction is unique up to isomorphism shouldn't I prove that as well? However continuing on with this fashion would bring me out to almost two weeks spent on the first few chapters of a text by Rudin.

Though I've sat in on about three intro to real analysis courses, taught by different instructors at a university and with the amount of time they each spend on some subjects e.g. at the start when discussing say completeness properties, right before jumping into metric spaces etc. my only conclusion is that essentially every undergraduate has taken much of these constructions on faith for the amount of time it seems required to do this in depth is no where near the few classes spent on the topic. Are people not expected to prove this stuff? How much time should I spend doing this? Its like each time I try to formalize something it opens up a new problem, when do I stop digging? Do I just accept this stuff like other students and move on to neighborhood Topologies etc. it seems were I actually enrolled in these classes I'd have no other alternative if I wanted to pass any exams.

Answer 1

"Should one prove" that? No, it's not necessary. It's already a very interesting, rich, and fruitful mathematical enterprise to learn the consequences of the axioms of complete ordered fields, and from a purely logical standpoint, that's exactly what real analysis is about.

On the other hand it's evident that you are very interested in knowing the proof of isomorphism, in which case I would encourage you to pursue the issue. The proof is not too hard, and is sometimes presented in a real analysis course (although I have not presented it in my course). Until you get around to studying that proof, you can just think of real analysis as the study of "a" Dedekind complete ordered field rather than "the" Dedekind complete ordered field; the theorems one proves will be true about any Dedekind complete ordered field.

But it's very, very important not to let this turn into a mental block which stops you from putting in the necessary time to study real analysis.

Answer 2

This is not an answer to your question, but in case you are interested, a way to prove that all Dedekind-complete ordered fields are isomorphic to $\Bbb R$ is:

• Starting with $0$ and $1$, use induction to embed $\Bbb N$ into your field.
• Extend that embedding to $\Bbb Z$ and $\Bbb Q$ arithmetically.
• Extend it to $\Bbb R$ by Dedekind cuts (or equivalent).
• Show that the image of $\Bbb R$ under the embedding cannot have a supremum. (If $\omega$ is the supremum, then $\omega - 1$ is less than some real number.)
• Show that if the field has an element $\epsilon$ not in the image of the embedding, then it has to have an element greater than any real number. (If $\epsilon$ and $-\epsilon$ are both not greater than all reals, then $\epsilon$ must lie between real numbers, so you can approximate it with reals. Let $\delta$ be the supremum of all such approximations that are $< \epsilon$. As a supremum of bounded reals, $\delta$ is real itself and so is not $\epsilon$. Now consider $1/(\epsilon - \delta)$.)
• Conclude that the field cannot be complete if it has such an element, as it has a set - the embedded $\Bbb R$ - which is bounded above, but has no supremum.

Hence the entire field is the embedded image of $\Bbb R$ and so is isomorphic to $\Bbb R$.

Answer 3

Well, I did not really study a proof that all Dedekind complete ordered fields are isomorphic. But I did study the construction of real numbers, first the one given by Dedekind and then the one given by Cantor (Cauchy sequences) and then showed that these are isomorphic just to get a feeling that these are essentially the same thing. Another thing which I did was to prove that all different versions of completeness of reals are equivalent. And this does help a lot in understanding the reals of real-analysis.

IMHO the isomorphism thing does not help as much in understanding the key ideas of real analysis as the different versions of completeness. Each version gives a different but correct answer to the question: how are the reals different from the rationals? But this is a personal opinion and perhaps some people may find studying the isomorphism thing rewarding enough.