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For a frame of reference, I'm an undergraduate in mathematics who has taken the introductory analysis series and the graduate level algebra sequence at my university.

In my analysis class, our book lists axioms describing the structure of the reals. This seemed unnatural to me, as we can often write definitions or lists of rules that no set actually satisfies. So the axiomatic approach doesn't reassure me that the thing we are discussing can actually exist.

Our teacher talked to us about dedekind cuts as a way of explicitly constructing reals, which seemed more useful to me.

In my modern algebra class we discussed completion of the rationals as a way to construct the real numbers.

But this leads me to wonder - How do we know that all of these approaches and constructions result in the same structure, namely $\mathbb{R}$? Also, are the real numbers the only complete, totally ordered field? If so, why?

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    $\begingroup$ Related: math.stackexchange.com/questions/269353/… $\endgroup$ – David Apr 22 '17 at 7:15
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    $\begingroup$ I would strongly encourage checking out the answer by Kahen here as well... math.stackexchange.com/questions/11923/… $\endgroup$ – David Apr 22 '17 at 7:27
  • $\begingroup$ The simple answer: it is the only one of itself there is. There are equivalent ways of defining it, but ultimately, there is only one of it. If you can show that some process leads to that thing, you know it's the same one the other processes lead to, too. $\endgroup$ – Aza Apr 22 '17 at 8:47
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    $\begingroup$ I would never miss an opportunity to cite this exploration: arxiv.org/abs/1204.4483 $\endgroup$ – Patrick Stevens Apr 22 '17 at 12:51
  • $\begingroup$ what do you mean by "structure"? algebraically we have one signature but maybe many structures and thus many models satisfying the sig. $\endgroup$ – user325466 Apr 23 '17 at 21:25
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To paint a more complete picture, you are right in that an axiomatization may very well have no model. An axiomatization is meaningless if nothing satisfies it. But if we can prove that there is a model, and the axioms are the only properties we care about, then we can happily work within the axiomatization to prove theorems about the structure that we have axiomatized. That is precisely what we are doing when we use an axiomatization of the real numbers to prove theorems about real numbers.

Now there are two common axiomatizations of the reals. One is a second-order theory including the completeness axiom (every subset of the reals with an upper bound has a least upper bound). By itself, this axiom is useless, because there is no axiom that asserts the existence of any set of reals at all! However, when we use this second-order theory we are always working outside the theory in the foundational system (such as ZFC set theory) where we do have axioms that allow construction of subsets of the reals. It is this particular axiomatization that has a unique model; there is a unique model of the second-order theory of the reals up to isomorphism. That implies immediately that all structures that you construct (like the Cauchy completion of the rationals or the Dedekind completion of the rationals) that satisfies this second-order axiomatization must be isomorphic to one another.

The core reason behind the uniqueness of complete ordered field is that any two such fields must contain an isomorphic copy of the rationals, and every element in each field cuts the rationals in the field into two parts, and the lower part has a least upper bound, and that different elements cut the rationals in different ways (by the Archimedean property that follows from the completeness property as well). This gives a one-to-one correspondence between the reals in one field to the reals in the other field. One can say that it is the rigidity and denseness of the rationals that is the key.

However, the other common axiomatization of the reals is the theory of real closed fields. Note that this axiomatization does not have a unique model. The computable reals form a countable real closed field, and satisfies every first-order sentence that $\mathbb{R}$ satisfies. It may be instructive to see the same phenomenon with the natural numbers, which form the unique model of the second-order Peano's axioms (which was his original formulation) up to isomorphism, while there are many non-isomorphic models of first-order PA. The distinction here between the first-order induction schema and the single second-order induction axiom must be appreciated for one to understand how the second-order theory can pin down the natural numbers unlike the first-order one. More specifically, second-order induction applies to every subset of the natural numbers (as seen from the foundational system), while first-order induction applies only to subsets that can be described using an arithmetical formula. There are uncountably many subsets, but only countably many formulae.

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    $\begingroup$ Most of your answers seem to have a unique property irrespective of the votes received. There is always a new viewpoint like the last paragraph here. I was familiar with the unique of complete ordered field but this one about real closed fields is something I need to look into. And even what you describe about complete ordered fields is very nice. I especially like the phrase "rigidity and denseness of the rationals is the key". +1 $\endgroup$ – Paramanand Singh Apr 22 '17 at 15:12
  • $\begingroup$ Wonderful post. Typo: Dedekind. Can't submit a one character edit. $\endgroup$ – chx Apr 22 '17 at 15:42
  • $\begingroup$ @chx: Haha I'm very amused I didn't notice that while typing; I usually don't make typographical errors. Thanks a lot! $\endgroup$ – user21820 Apr 22 '17 at 15:48
  • $\begingroup$ @ParamanandSingh: Thanks again for your kind comment! I'm glad my post has something interesting to you too! Real closed fields is intriguing because the computable reals satisfy it, so it can't have completeness; there is a set of computable reals with no computable supremum. But curiously, that's not saying much; after all "set" here is completely arbitrary; of course we could simply choose the set of all computable reals that are less than some arbitrary uncomputable number and there we have a counter-example to completeness. $\endgroup$ – user21820 Apr 22 '17 at 15:52
  • $\begingroup$ @ParamanandSingh: I just happened to come back here, and noticed that I missed an interesting bit from my post. Note that if our foundational system (such as ZFC) is consistent, then it has a countable model, inside which the reals appear uncountable (no internal bijection with $\mathbb{N}$) but are externally countable (because any object in a countable model of ZFC has only countably many members). Thus the categoricity of the reals (namely uniqueness up to isomorphism) is not at all absolute, and can only pin the reals down (essentially) within a single model of the foundational system. $\endgroup$ – user21820 Oct 6 '18 at 6:06
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The (fake1) history of mathematics is

  • We believed we understood the real numbers very well
  • We wrote down simple properties (we believed) that the real numbers satisfy
  • We checked that these properties are enough to actually prove from axioms all of the things we were proving about real numbers

Uniqueness, incidentally, comes from the third point. For example, one of those "things we were proving about real numbers" is that they are the completion of the rationals. Since the rationals are unique and completions are unique, the completion of the rationals must also be unique.

(that a complete ordered field contains the rationals follows from "field" and "ordered", and the topology on a complete ordered field follows from "ordered")

As usual, by "unique" I really mean "unique up to isomorphism". In this case, we even have "unique up to unique isomorphism".

1: true history is strange; it may not have happened in this order, or precisely like this

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    $\begingroup$ Note that there is also a fake prehistory of mathematics in which we thought that real numbers were rationals but then some cleverclogs drew the diagonal of a square and invited us to put it in our pipe and smoke it. $\endgroup$ – David Richerby Apr 23 '17 at 13:23
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As Pugh states in Real Mathematical Analysis, the real numbers are the unique, up to order-preserving isomorphism, complete ordered field containing the rational numbers as an ordered subfield. For suppose $\mathbb{F}$ is a complete ordered field containing $\mathbb{Q}$ as an ordered subfield. Then the map $y \mapsto \{ q \in \mathbb{Q} :q < y \text{ in } \mathbb{F}\}$ is a order-preserving isomorphism from $\mathbb{F}$ to $\mathbb{R}$, where $\mathbb{R}$ is viewed as a set of Dedekind cuts.

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A lot of books devoted to the development of the real number system (and other number systems) were published in the 1960s. During the past few years I've been keeping a list of such books (regardless of when they were published) when I happen to see one in a library, and the ones I've come across so far are listed below.

Note that if we expand the list to books in real analysis, "transition to advanced mathematics", abstract algebra, metric spaces and/or topology, etc. that include such a treatment, the list of such books would be at least several hundred. Thus, I've restricted the list below to only those books that are primarily devoted to this topic. The links I've provided are not necessarily to the same edition that I've given information about.

Besides what others have recommended, you can try visiting a nearby college or university library and look for some of these books. Some of these books will probably be more useful to you than others for what you want, and you will likely find other books in the same general shelving location that are not in the list below.

[1] Leon Warren Cohen and Gertrude Ehrlich, The Structure of the Real Number System, The University Series in Undergraduate Mathematics, D. Van Nostrand Company, 1963, viii + 116 pages.

[2] Solomon Feferman, The Number Systems. Foundations of Algebra and Analysis, Addison-Wesley Publishing Company, 1964, xii + 418 pages.

The 2nd edition was published by Chelsea Publishing Company in 1989 (xii + 418 pages).

[3] Norman Tyson Hamilton and Joseph Landin, Set Theory and the Structure of Arithmetic, Allyn and Bacon, 1961, xii + 264 pages.

[4] Edmund Jecheksel Landau, Foundations of Analysis, 1951, Chelsea Publishing Company, 1951, xiv + 134 pages.

Translation by Fritz Robert Steinhardt of the 1930 German edition (xiv + 134 pages).

[5] Elliott Mendelson, Number Systems and the Foundations of Analysis, Academic Press, 1973, xii + 358 pages.

Reprinted by Robert E. Krieger Publishing Company in 1985 (xii + 358 pages). Reprinted by Dover Publishers in 2008 (xii + 308 pages).

[6] John Meigs Hubbell Olmsted, The Real Number System, Appleton-Century Monographs in Mathematics, Appleton-Century-Crofts, 1962, xii + 216 pages.

[7] Francis Dunbar Parker, The Structure of Number Systems, Teachers. Mathematics Reference Series, Prentice-Hall, 1966, xiv + 137 pages.

[8] Joseph [Joe] Buffington Roberts, The Real Number System in an Algebraic Setting, A Series of Undergraduate Books in Mathematics, W. H. Freeman and Company, 1962, x + 145 pages.

[9] Hugh Ansfrid Thurston, The Number-System, Interscience Publishers, 1956, viii + 134 pages.

Reprinted (slightly corrected) by Dover Publications in 1967 and 2007.

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  • $\begingroup$ That's pretty thorough :-) You may be interested in the article Weiss, Ittay. Survey article: The real numbers—a survey of constructions. Rocky Mountain J. Math. 45 (2015), no. 3, 737–762 here. $\endgroup$ – Mikhail Katz Apr 24 '17 at 15:27
  • $\begingroup$ @Mikhail Katz: I posted a list myself back in 17 June 2006, but I'm sure the folder in which I put such papers when I come across them (the folder is at home somewhere) has a lot more such papers in it now. I'll have to look up the Rocky Mountain J. Math. paper the next time I'm at the library and stick it in my folder. $\endgroup$ – Dave L. Renfro Apr 24 '17 at 15:45
  • $\begingroup$ Thanks for providing this list. It will be helpful to me. $\endgroup$ – Eigenfield Apr 12 '18 at 11:15
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The uniqueness of the real numbers is a bit of a fantasy that reinforces Platonist ideas about mathematics, which may or may not be a bad thing. At any rate, consider the following thought experiment that should indicate that even the uniqueness of the natural numbers $\mathbb{N}$ is dubious:

Is there a set of cardinality strictly between $\mathbb N$ and $\mathbb{N}^{\mathbb N}$?

At a more technical level, it could be pointed out that the so-called categoricity of the real numbers implicitly depends on the choice of the background set theory. Change the background theory and you change the real numbers.

At a further technical level, in Edward Nelson's Internal Set Theory, infinitesimals can be found within the ordinary real numbers. Not every college freshman's idea of the real numbers, is it?

Thus, such "uniqueness" is dubious from at least two points of view: (1) the theory is not unique; and (2) the model is not unique.

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No and yes. It depends by what you mean by unique.

First, for the No side. Here are some examples of some questions in which the answer completely dependent on which $\mathbb{R}$ are you talking about (and were significant unsolved problems in analysis for a long time before people realize it cannot be answered with our current axioms):

  • Let $f$ be an arbitrary function that assign each real number to a countable subset of real numbers, can you always find 2 numbers $x,y$ such that $x\notin f(y)$ and $y\notin f(x)$.

  • Is there a subset of $\mathbb{R}^{2}$ such that every vertical line intersect the set at countably many points and every horizontal line intersect the set at uncountably many points.

Now, for the Yes side. It is a standard theorem in analysis is that any 2 ordered fields with the Dedekind complete property are isomorphic.

How are they both possible? When you talk about unique, you are talking about real number inside the same universe only. It say nothing about real number in a different universe. In particular, 2 "distinct" (e.g. they have different properties) ordered fields with Dedekind complete properties cannot lie in the same universe. However, it is entirely possible that 2 universes contain different real number.

People can't even agree as to which is truly the real number. People have better ideas about what natural number looks like, and even then, we can't even write down axioms that will pin down the natural numbers.

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    $\begingroup$ The reals are essentially unique in any universe of set theory. There is no such thing as "absolutely unique" anyway, because you need some kind of set theory to have the second-order axiomatization of the reals, and any first-order set theory has a countable model if it is consistent. And I don't see why you claim that the standard natural numbers are unique in your sense. Their definition in terms of finite strings is only possible in a strong enough system, but if consistent it has two models that disagree on what are the finite strings! So I've no idea what your point is. $\endgroup$ – user21820 Apr 22 '17 at 16:58
  • $\begingroup$ @user21820: as I mentioned in the answer, no axiomatic system can pin down the natural number, so it is not unique in the sense of axioms in the first place. But it is unique, because human know and distinguish between actual natural number and fake one. For example, (assuming ZFC is consistent), there is an universe where $\neg Con(ZFC)$ is true, in other word there is a "natural number" which encode the "proof" of inconsistency of ZFC. But any reasonable person will agree that such "proof" is not real, and such "natural number" is fake. $\endgroup$ – carrotomato Apr 22 '17 at 17:15
  • $\begingroup$ Ok I get your point. However I disagree with the premise that "humans know and [can] distinguish between actual natural number[s] and fake one[s]". The reason is that you only have evidence that we can identify very tiny natural numbers (surely restricted to less than $10^{20}$ bits). No human has access to all the natural numbers, so one can't really say that humans can distinguish between all natural numbers (in some physical representation) and all other objects... In any case, you are not actually answering the question; this is Math SE and not Phil SE. Hope you understand. $\endgroup$ – user21820 Apr 23 '17 at 6:03
  • $\begingroup$ @carrotomato, while I agree with the main thrust of your answer (and have upvoted it), it could be toned down somewhat, for example by deleting the first line. Make your point and let the reader decide if they agree. $\endgroup$ – Mikhail Katz Apr 23 '17 at 10:44
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    $\begingroup$ @carrotomato I sense that the demographic on the streets you frequent is somewhat different to the ones I know. I'd expect the answer to be, "You what?" As such, I find it very hard to be convinced of anything by your man-on-the-street argument. $\endgroup$ – David Richerby Apr 23 '17 at 20:16
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A real number system is a tuple (R,+,.,< ) which is an ordered field which is order complete (or instead of order complete there are several equivalent properties ) . Any two such systems are indeed isomorphic ,meaning there is a 1-1 correspondence preserving the structure ,all this has been already pointed out by others . What nobody seems to have said and which is a MAIN POINT (pun intended) ,is that this isomorphism is UNIQUE ; there is no choice of which elements correspond . In particular an isomorphism of (R,+,.,<) with itself must be the identity function taking x to I(x)=x . This is very different than say for 3 dimensional vector spaces ,any 2 have many vector space isomorphisms . This uniqueness result explains why the particular constructions ,Dedekind cuts ;Cauchy sequences and others ;don't matter much to analysis ;they do matter to people interested in foundations of almost all of mathematics within set theory from set theory axioms (me included ) but not to analysts (me included) .Regards,Stuart

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  • $\begingroup$ You're right that the isomorphism is unique, but I don't think it's the key issue here. The question was about uniqueness up to isomorphism, but does it really matter whether the isomorphism is unique? The complex numbers is the unique algebraic closure of the reals up to isomorphism, but the isomorphism is certainly not unique. Yet it is incredibly important, and both the complex analytic construction and field theory construction are completely different (with fundamentally different proofs) and yield essentially the same algebraic closure even though there are uncountably many isomorphisms! $\endgroup$ – user21820 Apr 24 '17 at 11:33
  • $\begingroup$ If ( C,+,. ,i, ) is a fixed complex number system containing a real number system R with + and . restricted to R as the R-operations then of course if we require a ring isomorphism g to fix the elemennts of R andg( i)=i ( where I^2=-1 ) ., then of course the isomorphism is trivially the identity . If one doesn't insist that g(I)=i ,the conjugation x+iy --> x-iy (x,y real ) is also a ring isomorphism but I think there is a very hard theorem that says thats all you can have . $\endgroup$ – StuartMN Apr 24 '17 at 23:04
  • $\begingroup$ Um that theorem is not hard at all; trivial because C is merely a quadratic extension of R. However, you are fixing the copy of R inside C, and even then you have an arbitrary choice of $i$. This is unlike the categoricity of the second-order theory of the real numbers. $\endgroup$ – user21820 Apr 25 '17 at 1:49
  • $\begingroup$ Thanks for the comments. what does categoricity mean ? $\endgroup$ – StuartMN Apr 25 '17 at 9:38
  • $\begingroup$ A theory is called categorical iff it is satisfied by a unique structure up to isomorphism. It is a key theorem of logic that every first-order theory with an infinite model is not categorical, because it would have a model of every infinite cardinality. Thus no first-order theory can hope to be categorical. It is still possible to be κ-categorical, meaning there is an essentially unique model of size κ. But note that (first-order) PA is not $\aleph_0$-categorical, and RCF (as mentioned in my answer) is not $\#(\mathbb{R})$-categorical either. $\endgroup$ – user21820 Apr 25 '17 at 9:45

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