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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.

user3865391
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    Mathematics is full of black boxes. While I think that it is admirable that you are trying to unpack one of them, it might be better for you (as a student) to focus on the classes that you are meant to be taking. – Xander Henderson Jan 17 '18 at 14:45
  • @XanderHenderson I'm not a student, I have access to a mirrored version of course listing and typically just walk in on the classes and am quiet so no one bothers me. Though with regards to the question, should I just accept some stuff at faith, and then go back later and prove it in the future? I mean it seems this is what most students do with mathematics as I've talked to engineering students working on partial differential equations among other what I think to be fairly advanced topics that didn't know naive set theory. – user3865391 Jan 17 '18 at 14:56
  • There are thousands of problems, theorems, and proofs about real numbers which you don't have seen and studied yet; but you are welcome to do useful work with these numbers anyway. A doctor is able to cure his patients without a clear knowledge about the fine structure of matter. – Christian Blatter Jan 17 '18 at 18:56

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"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.

Lee Mosher
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  • So its okay to assume some stuff and then later go back and prove it when I have the tools, like build part of my house then work on the basement (sorry if that's a bad analogy). – user3865391 Jan 17 '18 at 15:00
  • Think of it this way: to prove any theorem which has the format "If P then Q" you have to assume some stuff (namely P). When you wish to go back later and apply that theorem, yes, then you'll have to prove that P is true when you have the tools. Meanwhile, the theorem "If P then Q" remains a true theorem ready to be applied. – Lee Mosher Jan 17 '18 at 15:03
  • Sorry so basically you're saying yes then right? For the introduction don't spend $30$+ hours trying to prove/construct all the formal stuff, I mean obviously still prove theorems throughout the reading but for the big formal stuff like that it would be okay for me (in your opinion as a mathematician) to hold it off until later when I'm more knowledgeable? Also again thanks for your time I upvoted, just want to be clear before I continue on reading and not spending time with this anymore. – user3865391 Jan 17 '18 at 15:11
  • I'm not sure, really, what you mean by the "formal stuff". I'm saying "Yes, it is not necessary to prove that Dedekind complete ordered fields are all isomorphic to each other in order to study real analysis". In addition, I'm trying to give you some guide on how to sensibly work your way through learning mathematics. – Lee Mosher Jan 17 '18 at 15:34
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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 $\Bbb R$.

Paul Sinclair
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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.

Paramanand Singh
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