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Here is my question. Why do the reals need to be "constructed" by this bizarre "Dedekind cut" or "equivalence class of Cauchy sequences" argument? Why can't they simply be "observed" as consisting of all numbers that "span" some known sets of numbers?

I am thinking here, in part, by analogy with linear algebra and with the complex numbers, where $i$, the square root of $-1$, is really all you need, in addition to the reals, to get all complex numbers as a spanning set of $1$ and $i$ over $\mathbb R$. (Every complex number may be expressed as $a\cdot 1 + b\cdot i$ where $a, b\in\mathbb R$.)

We have a couple of known transcendental numbers, $e$ and $\pi$. We have all the rationals. We have all the square roots most of which are irrational. We have all the fractional roots of $e$ and $\pi$. We have the $e$th roots of all the numbers that exist, and the $\pi$th roots. Maybe we also have some other sets of transcendental numbers out there that we can use?

What I am trying to ask is, are we using these "Dedekind cuts" and "equivalence classes of Cauchy sequences" just because we don't "know enough real numbers yet", because their characterization hasn't occurred to us yet, or do we already have enough real numbers in our arsenal, like e and pi, to make a "spanning set" without using equivalence classes of infinite sequences and the like, or, is it the case that we really have to use these kinds of constructions of the reals, for some deep mathematical reason?

It just doesn't seem right. Because you have to admit, by identifying "sets of numbers" like Dedekind cuts and equivalent classes of Cauchy sequences which are both sets of numbers, with actual numbers, mathematicians create (at least in my mind) some cause for doubt about what they are doing here with the reals. A "set" seems like a strangely undefined term, which I understand, but not well, is subject to various kinds of paradoxes and levels of analysis problems. (This last paragraph may be more of a separate question, about the validity of using sets of numbers as numbers, from the first question, which is more about why aren't there simpler ways to define or understand the real numbers in terms of numbers and operations we already understand.)

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There are way more reals ($\beth_1$, continuity) than constructible reals (mere $\beth_0$, enumerable). –  Jan Dvorak Jan 16 '13 at 7:21
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In math, we don't "observe", we prove. As a matter of fact, already the natural numbes are usually constructed set-theoretically in such a way that each number is in fact a set of numbers. –  Hagen von Eitzen Jan 16 '13 at 7:22
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The point of integers is to close $\mathbb{N}$ under subtraction. The point of the rationals is to close $\mathbb{Z}$ under division (as best as possible). The point of the reals is to close $\mathbb{Q}$ under limits. Limits are not an algebraic sort of operation, and are much more closely tied to converging sequences and cutting a line than they are to any algebraic operation. –  Robert Mastragostino Jan 16 '13 at 7:33
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Also set are perfectly well-defined, though your teacher may not have been so thorough about it because the subject can get hairy quickly. And the integers and rationals are often constructed by equivalence classes from the naturals, so I'm not sure what should be so bizarre about doing this one more time. Sets are better defined than numbers are. If you think using sets to define numbers is odd, I think the biggest issue with resolving that issue would be deciding on what a number exactly is. –  Robert Mastragostino Jan 16 '13 at 7:35
    
@Robert Mastragostino, I wish you would expand your comment about $\mathbb{R}$ being the closure of $\mathbb{Q}$ under limits into an answer. It gets exactly at the OP's original question as to whether or not we already have "enough" numbers without the reals. To do analysis, we don't. You should get the credit for raising this point. –  trb456 Jan 16 '13 at 21:02

8 Answers 8

up vote 19 down vote accepted

If you like, and some people do, you can forget about any construction of the reals from the rationals (or anything else) and instead define them axiomatically. One such axiomatization is Tarski's.

This approach will avoid any weird feeling you might have about a real number being an equivalence class of whatnot.

Usually, the reason to provide an explicit construction of something from a simpler things is that it proves that that something exists (mathematically). Moreover, it allows you to study properties of that something in terms of the simpler things that you presumably know better.

Nobody things of real numbers as equivalence classes of anything. Once the construction is done you can just forget about it if you like. Having a construction just means that the model of the real numbers that you fantasize about is at least as consistent as a model you might have of the simpler things. To some people it gives reassurance, to others a headache.

As for your attempt to define the read as something spanned by those things we have names for, together with some operations on there. The problem is that there are only countably many such things while there are uncountably many real numbers (at least if you believe that every real numbers admits at most two decimal representations). So this can't work. It might be strange to think about there being more reals then potential names or ways to approximate reals but it's a real fact (pardon the pun).

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Many thanks for the suggestion to look at Tarski. I think for me, if I understand correctly what this is, replacing one set of axioms (axioms defining a well-ordered field) with another, smaller set of axioms (Tarski), doesn't do much. –  user58450 Jan 16 '13 at 8:51
    
Someone please add an answer that refers to "least upper bound", thanks! –  barrycarter Jan 16 '13 at 18:35
    
Something is still bugging me about all this and that is, aren't e and Pi counterexamples! Pi is not an equivalence class of Cauchy sequences. It is not a Dedekind cut. It is a real number. Obviously, it is intrinsic to the relationship between the length of the radius and the area (and circumference) of the circle. So: can real numbers be fully classified by their geometric relationships or implications, and if so, do those geometric relationships or implications form a basis for describing the reals that is better than using the mere property "every real number gets converged to"? –  user58450 Jan 18 '13 at 23:22
    
@user58450 in a construction of the real numbers using equivalence class of Cauchy sequences the numbers pi and e will both be represented as equivalence classes. In a construction using Dedekind cuts they will be represented by Dedekind cuts. "... can real numbers be fully classified by their geometric relationships and implications..."? No! –  Ittay Weiss Jan 18 '13 at 23:40
    
Many thanks for your answer. Obviously still thinking this through and all these responses really help. –  user58450 Jan 18 '13 at 23:48

To be honest, I find this question quite obscure - yet there have been a lot of very detailed, deep answers discussing different approaches such as Tarski's axiomatic introduction of the reals etc.

However, in my opinion, there is one very simple point about the question that is - mathematically speaking - incorrect and resolving this, we receive a much simpler answer.

We have a couple of known transcendental numbers, e and π. We have all the rationals. We have all the square roots most of which are irrational. We have all the fractional roots of e and π. We have the eth roots of all the numbers that exist, and the πth roots.

Now, as long as we're dealing with the set of rational numbers, we do not "have" any of these. We do not even have the vocabulary to say in any sensible way "the number $\pi$", because $\pi$ is not a number as long as we are "inside" $\mathbb{Q}$ . Neither could you say $3.1415926...$ since the limit process hidden behind the "..." does not have a limit (once again, in $\mathbb{Q}$). You might say "well, I'll just define it as the limit of this Cauchy-sequence and add it to $\mathbb{Q}$" - but then it is highly doubtful what you mean by the limit of a sequence that does not converge. The limit in a larger set? - you would need the reals in the first place!

You might want to take the expression "limit" purely formally. This will then identify your "number" $\pi$ with the sequence you chose to approximate it. For consistency reasons, you might want your "number" to be independent of the sequence you chose (remember that, for the limit, you can always forget about the first few numbers in the sequence) and you will quite naturally end up doing exactly the usual completion process using equivalence classes of Cauchy-sequences.

So, if you are talking about "the numbers $\pi$ and $e$", I kindly ask you to clarify what you mean by these expressions. And I am quite sure that you will actually run into either of these constructions. And if you find a new construction method - even better!

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A very good question, in part because it will be hard to say whether an answer is the right answer. There are many good answers here already, many correct answers. But the question has interesting ambiguities, whether intended or not. Here I offer a few things, from my own perspective, which I think will contribute something to the set of answers.

First, "need." In an extreme sense, we don't even need all the integers. All the calculation we need to make things can be done on a finite machine. A slightly less extreme position is that the computations in making things have always been done with finite precision. I am not being facetious, because at some point the need comes down to what one is willing to assume just works and what one is unwilling to accept without justification. Many people are willing to assume that there are numbers that work just fine, and they go out and design and build bridges and financial systems and so forth. This sort of choice happens in mathematics as well. One may choose to work from a set of assumptions while another may choose to investigate how those assumptions may be justified. While everyone recognizes the importance of sound footing, they also realize there are important problems someone should think about and not wait on the highly unlikely outcome that there are fundamental problems with our assumptions.

Second, there is the question of whether such a manner of construction of the reals is necessary. The suggestion that the real numbers are "all the numbers that span a certain something" presupposes that some numbers have been constructed or otherwise exist; and among those numbers, some have a property that would distinguish them as real. The supposition raises the question of how these other numbers came to exist, were they constructed or are they assumed to exist. It would be simpler to construct the reals directly via cuts or sequences or, as has been suggested, to assume they exist with the necessary properties.

Third, to do analysis, it is not strictly necessary to construct the reals, for one could construct the surreal numbers instead. They "work" (that is, could be used to do analysis), since they contain a field isomorphic to the reals. But the reals tend to be more convenient.

Fourth, there is the question of the purpose or a "deep mathematical reason" in constructing the reals. Someone summarized one reason very nicely in a comment: for the sake of limits. There is an older reason. The basic idea in Dedekind's development of irrationals may be found in Euclid (Bk. V, Def. 5.). It is attributed to Eudoxus. The purpose was to develop a rigorous theory of similar figures that could handle irrational proportions. Since the concept of a real (or even a rational) number was absent, a definition in terms of integral multiples was needed. With it, Euclid proved the gem of Bk. VI, a generalization of the Pythagorean theorem, that if similar figures be constructed on the sides of a right triangle, the area on the hypotenuse equals the sum of areas on the sides.

Finally, another purpose was to develop calculus without appealing to any intuitive notions of what a geometrical line is like and or what an inifinitesimal is. Sometimes one might "see" something a certain way but it turns out not to be that way. Something like if $f'(0)=10$ then $f$ is increasing near $0$. It turns out not to be true always, even though one sometimes (loosely) talks of the graph having an upward slope. Such loose thinking might produce errors in analysis, and some thought it was important to show that the analysis of real-valued functions of real variables can be founded on a theory that depends only on numbers and not on geometric properties.

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Michael Ex on your comment about the nonstandard reals -- I did take a look at these briefly. I don't think they have actually constructed the "nonstandard reals" at all. Instead, what they appear to have done is to tack on their theory of infinitesimals, to the existing reals which they assume to exist without question and without any detailed definition or construction. Thus, these guys get away with neatly bypassing almost all the issues in my question about R. [There is a character limit to my comment so I am going to finish this response with additional comment.] –  user58450 Jan 18 '13 at 22:13
    
If they were really "constructing the nonstandard reals" then they would say something like "assume the rationals" and add to them, numbers infinitesimally close to the rationals. They do NOT do this! Instead, they say "assume the reals" (!) for example citing the free text Elementary Calculus: an Infinitesimal Approach by Keisler online at math.wisc.edu/~keisler/calc.html page 27: "The real numbers form a subset of the hyperreal numbers..." They use the reals to define the hyperreal extension of R. Perhaps an awesome way to do calculus but so far not helpful to me re R. –  user58450 Jan 18 '13 at 22:21
    
These nonstandard analysis guys I think actually explain why somebody can't do a "nonstandard analysis" version of the reals that assumes only the rationals and augments them with numbers "infinitely close" to the rationals. Their reason is that if you pick any real number, it is demonstrably not "infinitely close" to any particular rational number whatsoever. In other words, that idea doesn't work. But this whole thing with R still really bugs me; I just really dislike this "infinite sequences (or infinite sets in the case of Dedekind) of numbers" definition of R. –  user58450 Jan 18 '13 at 22:31
    
I will post back if/when after further thinking, I feel I really honestly "get" why this Cauchy/Dedekind thing is such an absolutely necessity. I do see how there is this need for the "continuum" to be proven to "exist" so that there aren't any "holes", and I sort of see how it is helpful to actually be able to say "please refer to exhibit A, your Honor: equivalence classes of Cauchy sequences! These are the same thing as this ordered continuum that we are trying to prove exists, so that means that this ordered continuum thing exists." All these comments have been helpful. –  user58450 Jan 19 '13 at 0:40
    
@user58450 Thank you for your comment(s). My point was not, I think, of terrible significance: if B suffices instead of A, then A is not, strictly, logically necessary (B = nonstandard $R$, A = standard $R$). Nonetheless, standard $R$ is the traditional foundation of analysis and the question of its construction retains importance. Who are "they" you refer to? You seem to have a notion of "nonstandard $R$" that is anterior to a definition of it. Surely, Robinson is free to define or construct a set ("nonstandard" $R$) as he pleases. That he did so successfully is generally acknowledged. –  Michael E2 Jan 19 '13 at 1:50

Lots of people have given good answers, but they seem to me not to be hitting your main question. You asked why the reals have to be constructed, and is the reason perhaps that there are "not enough" numbers. The answer is, if you are doing analysis, then no, there are not enough numbers, and the constructions you currently find bizarre are actually the way the need for the reals arise.

Say you just want to use the rational numbers. They're a nice, ordered field, and they are the space where we in essence take all "real" measurements, like in science. But in analysis, and its more well-known application calculus, the main new tool one uses is limit-taking: approaching points infinitesimally. Suppose you say you'll only approach points "nicely", with sequences that never diverge off to infinity, don't oscillate wildly, etc. A great class of "nice" sequences are the Cauchy sequences, where the points get "arbitrarily" close together the further out in the sequence you get.

Unfortunately, this idea won't work. Look at this sequence:

$3, 3.1, 3.14, 3.141, 3.1415,...$

It's monotone, bounded above and below, and so Cauchy, and it consists only of rationals, but it is obviously meant to converge to $\pi$. So the logical thing to ask is: what is the smallest number of additional points I need to add to the rationals to make it so all rational Cauchy sequences converge? But that's the whole real line.

This is why things like the computable numbers seem appealing, and may be needed in explicitly constructive applications. But nothing short of the reals guarantees rational Cauchy convergence, and this is such a basic need in analysis that anything less is simply a huge headache; i.e. you'd have to condition any theorem involving limits on whether or not the limit exists in your chosen space that is not the whole real line.

I think most of the other answers here are better than this one, and give more interesting details, but I did not want this main point to slip.

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But what if I only demand that computable Cauchy sequences converge? Even with the Dedekind cut construction, there is a subtlety about what kinds of subsets "exist"... –  Zhen Lin Jan 17 '13 at 23:43
    
I just checked the Wikipedia article on computable numbers. Note that all rationals are computable, so your idea won't work. But if you take computable sequences with a computable modulus of convergence (i.e. the rate of convergence is also computable) in addition, then you get computable analysis, which seems similar to constructive real analysis. But this is a very different animal, quite different from classical analysis (e.g. equality is not computable; all computable real functions are continuous). –  trb456 Jan 17 '13 at 23:53
    
Each rational number is computable, but it is not at all obvious to me that a Cauchy sequence of rational numbers is computable. –  Zhen Lin Jan 18 '13 at 0:01
    
It's not: the set of all limits of rational Cauchy sequences is the reals. But if you take a sequence of computable numbers, and add the requirement that the modulus of convergence also be computable, you get computable limits. So this means that there are fewer such sequences: i.e. there must exist sequences of computable numbers which converge "normally" (real modulus of convergence) but don't have a computable modulus of convergence. –  trb456 Jan 18 '13 at 0:12

You can actually do math without explicitly constructing the real numbers, although you end up constructing them implicitly.

If you accept that the number "1" exists, as well as the basic operations plus, minus, multiply, and divide, you can construct the rationals.

Even though you can't picture fractions like 3559/3571 in your head, you can certainly see how they could be constructed.

Sadly, there are several problems you can't solve with rational numbers:

  • x = x + 7
  • x*0 = 5
  • x^2 = 2
  • x^2 = -1

Why does not having a solution to "x^2=2" bother us more than not having a solution to the other problems above?

Answer: you can find rational numbers p and q such that p^2 < 2 and q^2 > 2 AND make q-p < epsilon, for any rational value of epsilon, no matter how small.

In other words, you can "squeeze" rational number squares as close to 2 as you want, without actually touching it. This offends our intuition, although Zeno claims it's quite normal (we try to punch him, but can get only arbitrarily close).

How do we solve this problem? Several possibilities:

  • Create a new number "s" and declare that s^2=2. Of course, this doesn't help with problems like "x^2=3" or "x^3=2".

  • Declare that every polynomial with rational coefficients is now also a number, namely the number that solves the polynomial itself.[1]

  • This seems to work fairly well, until someone points out the ratio of a circle's circumference to its diameter is not part of your number system. Again, you can arbitrarily close to that ratio, but never quite hit it.

  • So, how do you solve this new problem? You declare every set of rational numbers to be a number. Notice that you still haven't explicitly constructed the real numbers: your number system consists only of rational numbers and sets of rational numbers (we throw out the solutions to polynomial equations with rational coefficients since it's redundant).

  • With a little cleverness, you can define rules for adding, subtracting, multiplying, and dividing these new numbers you've created, both with each other, and with the rational numbers themselves.

  • How do these new numbers (ie, arbitrary sets of rationals) solve "x^2=2" and similar problems? You declare that a set S of rational numbers is a solution to f(x) = y, provided that:

    • For all r in S, f(r) <= y

    • For any rational epsilon, there exists r in S such that |f(r)-y| < epsilon

  • You have now implicitly constructed the reals, simply using rational numbers and sets. No real numbers anywhere.

  • Of course, there are a few problems with declaring any set of rationals to be a number. For example "x^2=2" now has an infinite number of solutions.

  • At this point, you might want to declare two sets to be equivalent under certain conditions (eg, the "least upper bound" condition), but this isn't really necessary: if you're OK with having infinite solutions to problems like "x^2=2", you can stop here.

There you have it: mathematics without explicitly constructing the real numbers!

DISCLAIMER: I realize this probably has some errors (eg, removing unbounded rational sets), and I intend it solely as a general guideline.

[1] In traditional mathematics, polynomials have multiple solutions, so declaring a polynomial to be a single number is admittedly a bit odd. However, I'm using this as a throwaway example.

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Constructing the reals is important if you want to do analysis. If you want to talk meaningfully about sequences or continuity, you need to fill in the "holes" in your space. You're coming from the perspective that we built the reals because we need "more stuff", but that's not the case. The reals are designed to fit together a certain way, and it just so happens that you need a lot of stuff to do that. If all the interesting analysis we wanted to do could be done with a smaller, countably infinite structure, it's possible that's what we'd call "the real line". In fact, I think some people do try and do analysis with the computable numbers.

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All the comments are great and much appreciated. Thanks for your suggestion about computable numbers. I just took a look at a free copy online of a book on computable numbers and I must say, the theory looks really interesting! Found it at the bottom of the list of references in the Wikipedia article on the subject. Link is here: eccc.hpi-web.de/static/books/… –  user58450 Jan 16 '13 at 8:44
    
Related to the idea of analysis with computable numbers: Tim Gowers has written a short dialogue on why we need the real numbers. It can be found here: dpmms.cam.ac.uk/~wtg10/reals.html –  kahen Jan 16 '13 at 21:38

You can get away as follows: You demand axiomatically that there exists a complete ordered field. It can be shown that any two such fields are canonically isomorphic and thus whatever someone assumes to be his personal idea or mental representation of $\mathbb R$, it is essentially the same as other people's idea as long as they agree to talk about a complete ordered field. (You than rather obtain $\mathbb Q$, $\mathbb Z$, $\mathbb N$ as subsets instead of constructing the other way round)

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You can read more about constructing the naturals, integers and rationals here. Also you can read Kahen's magnificent answer about showing that $\mathbb R$ is the only complete field here. –  JSchlather Jan 16 '13 at 7:41
    
Many thanks for your link. It will take some time to digest but I really appreciate that someone went to such lengths on this site to explain something!! –  user58450 Jan 16 '13 at 8:53
    
But there are many complete ordered fields not isomorphic to $\Bbb R.$ In order to characterize $\Bbb R$ one needs to further require the Archimedean property (or some equivalent). –  Math Gems Jan 16 '13 at 17:49
    
@MathGems Ah, it depends on what one understands as complete. Yeah, one better say complete Archimidean ordered field. –  Hagen von Eitzen Jan 16 '13 at 19:58
    
Thank you for the kind words, @JacobSchlather –  kahen Jan 16 '13 at 21:16

The vector space of $\mathbb R$ over the field $\mathbb Q$ is an infinite dimensional vector space. The reason is that $\mathbb Q$ is a countable set therefore $\mathbb Q^n$ is also countable, but $\mathbb R$ is not countable. So, we will need an uncountable basis to constuct real numbers from rational numbers.

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It should be added that not just any infinite orthogonal set will do as a basis. For example, the span of $e,\pi$ and the square roots of all positive integers over the field $\mathbb Q$ is not the whole set $\mathbb R$. In fact, it is impossible to explicitly give a basis without the axiom of choice; see math.stackexchange.com/questions/112859/… . –  Samuel Jan 16 '13 at 7:37
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Yes, I need to think through and read about this issue of how big R is, a bit more. It seems that all the commenters, though each may suggest a slightly different way of looking at the question, are comfortable with the work that has been done to date to construct or describe R. In other words, very educated commenters seem satisfied that this formalization for R is both quite solid and, perhaps, as useful, or comprehensible, as it needs to be to describe the subject. At some level, if it's not going to impact any practical problems in science and technology, then maybe it doesn't matter. –  user58450 Jan 16 '13 at 9:10
    
@user58450 solid certainly; as for useful or comprehensible - well, as Ittay Weiss' answer has it, "Once the construction is done you can just forget about it if you like" –  AakashM Jan 16 '13 at 12:19
    
@user58450 I think 'comfortable with the work to describe $\mathbb{R}$' actually depends a lot on who you're talking to. To analysts (those who care about the elements of $\mathbb{R}$ as numbers that can be added, multiplied, etc.) it's a comfortable, familiar place; but for set theorists and some topologists (who tend to think of it as 'subsets of $\mathbb{N}$' or 'functions from $\mathbb{N} \to \mathbb{N}$'), $\mathbb{R}$ is a scary, wild place where almost anything can happen. –  Steven Stadnicki Jan 16 '13 at 22:06

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