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I've been studying a collection of analysis books (one of them Bishop's Constructive version) and contemplating the reals. Correct me if I'm wrong, but I feel that I have seen the Cauchy sequence itself in some places and its limit in other places described as the real number. I do understand that nested closed intervals have a point as the limit of their countably infinite intersection. I can visualize an arbitrarily small interval of rationals, any of which has an equal claim (its seem to me, at least until other considerations are brought in) to being an "approximation" of this point. Unless we know the limit point (via the geometric series, for instance), what are we approximating if not a yet better approximation of a yet better approximation? (Maybe the "approximation" terminology works better for cuts.) I suppose I'm asking not only for the clarification of the dominant convention but also for "real talk" about the mathematical imagining of real numbers.

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There is a need of a formal definition of real number, though presumably in solving a problem, everyone or almost everyone starts from an intuitive notion. Depending on the formal definition one adopts, a real number is an equivalence class of Cauchy sequences, or a cut in the rationals. (These are the most popular, there are others.) I do not think anyone defines a real number as a Cauchy sequence of rationals, after all there are infinitely many Cauchy sequences that, intuitively, produce the same real number. –  André Nicolas Jun 5 '13 at 3:48
    
It's can't be the sequence itself, or else $\{1, 1.4, 1.41, 1.414, 1.4142,\ldots\}$ and $\{2, 1.5, 1.42, 1.415, 1.4143,\ldots\}$ would be different real numbers. –  MJD Jun 5 '13 at 3:55
    
Bishop likes to drop the equivalence classes and define every Cauchy sequence as a real number, but then defines equality in a similar way (more or less the same.) He definitely writes that the sequence is the number also. I see the need for a formal definition, absolutely. Along the same lines, I value algorithmic/constructive approaches precisely because they are mechanized, exact. –  s.z. Jun 5 '13 at 4:09
    
Or what. ${}{}{}{}$ –  Mariano Suárez-Alvarez Jun 5 '13 at 4:17
    
That's just it. I experience something like a clash of intuitive notions. The curve x^2 does not intersect the curve 2, right? We connect the imaginary dots when we plot functions, but most of the real points are limits of sequences we don't know the limits of, right? Or if they are equivalence classes, they are something like the mutual limit of equivalent sequences. I suppose that, for the counting-discrete part of my intuition, the real number is never finished, though in some cases we can compute the limit of an infinite series, for instance. –  s.z. Jun 5 '13 at 4:23

3 Answers 3

Real numbers are the "equivalence class" of Cauchy sequences of rational numbers. You take the set of all Cauchy sequences of rational numbers and put an equivalence relation on it. When do you say two Cauchy sequences of rationals say $x_{n}$ and $y_{n}$ are related? They are related if for every given $\epsilon > 0$ ($\epsilon$ is rational here), there exists a $n_{0}$ such that $n > n_{0}$ implies

$$ |x_{n} - y_{n}| < \epsilon $$.

Now you say that each equivalence class is a real number. How do you see the reational numbers inside real numbers then? So, suppose $r$ is a rational number, then the constant sequence $x_{n} = r$ is a Cauchy sequence and the equivalence class it belongs to is the rational number $r$ in the set of real numbers. This is the identification of rationals inside the reals.

This is one way of thinking about the real numbers. The other way is through Dedekind Cuts. You will find a wonderful exposition to Dedekind cuts in Walter Rudin Principles of Mathematical Analysis.

A yet another axiomatic way to think about the real numbers is that it is a complete ordered field, the natural numbers is the smallest inductive subset of real numbers, the integers are the subroup generated by natural numbers, and the rationals are the field of fraction of integers.

Pick and choose which you like. Any further discussion is welcome.

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Thanks. I do have Rudin and I looked into cuts as well. What I didn't find in Rudin is examples of the multiplication of cuts. It was all on such a painfully abstract level, aesthetically pleasing perhaps but like digging a hole with a spoon when wants an intuitive/metaphorical grip first. I know that the "cut" can be thought of as the lower set, for instance, but then you have a giant set from negative infinity to the cut as a real number. I believe that it works, but where is the concept there? An isomorphism, I suppose, but that's not satisfying. –  s.z. Jun 5 '13 at 4:06
    
@MJD thanks for the edit. –  Vishal Jun 5 '13 at 6:55
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@s.z. The intuition is not so complicated. The idea is that a real number, say $\sqrt2$, divides the rationals into two sets: the rationals that are bigger and the rationals that are smaller. And each real does this in a different way. So you let the division itself stand in for the real number. If all you have is the rationals, you can still specify a real value by cutting the rationals into a greater set (say, $\{x\mid x^2>2\}$) and a lesser set ($\{x \mid x^2 < 2\}$) and saying that the real value you are talking about is the one that divides the rationals that way. –  MJD Jun 5 '13 at 13:49
    
That makes sense to me. I find the basic notion of a cut intuitive. I suppose I would like to see some examples of cut multiplication, which I'm sure I could look up. I just find it a little frustrating that the books I own don't do that to drive the concept home. Seems to me that abstractions are "cooked up" in the mathematical imagination via exposure to particulars. –  s.z. Jun 5 '13 at 19:50
    
@s.z. cut multiplication (and division) is tricky when working with negative cuts. The easiest fix I've found for dealing with that is to construct "$[0,\infty)$" using cuts. Here multiplication of two cuts $\alpha$ and $\beta$ is easy: $\alpha\beta = \{ab \in \mathbb Q \mid a \in \alpha, b \in \beta\}$. It's a nice semifield and once its properties has been established, we can then mimic the construction of $\mathbb Z$ from $\mathbb N$ to produce $\mathbb R$ from "$[0,\infty)$". –  kahen Aug 21 '13 at 4:22

We have this Platonic ideal in mind when we think of the real numbers: Numbers on a line, densely ordered like the rationals, and like a line it doesn't have "any holes", i.e. some kind of completeness property (see the article Real Analysis in Reverse by James Propp).

It's easy to show that any two ordered fields, $R$ and $R'$, with the least upper bound property, say, are uniquely isomorphic, i.e. there exists a unique order-preserving field isomorphism $\varphi: R\to R'$. This guarantees that if we only use that the reals are an ordered field with the least upper bound property, then any statement we make using only those facts will be true for any set representation that has those properties as well.

In short: It doesn't really matter. All that matters is that there is some set representation and you're free to pick whichever one you like the best. I'm personally partial to the "equivalence classes of rational Cauchy sequences" and I've outlined a proof of existence and uniqueness in another answer of mine.

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Thanks. I suppose the completeness property formalizes our intuition of continuity. On the other hand, I think our notion of continuity clashes with our notion of the discrete (counting). Anyway, calling the real number an equivalence class of c.s. is still vague IMO when it comes to using that real number in the presence of rationals. Perhaps that's what disturbs me. Use is not well defined. Also the question of what is being approximated? A real number is a set, a sequence, a limit, for instance, but how, exactly, does that connect to use? –  s.z. Jun 5 '13 at 4:17
    
I have some extra reading for you: math.stackexchange.com/questions/119763/… and math.stackexchange.com/questions/261074/… –  kahen Jun 5 '13 at 4:26
    
I guess it's hard for me to believe that the number line has no holes. It's as if the irrationals are visualized as if already created, which makes sense in the concept of the unit diagonal but maybe not in the light of the positive integers as a foundation --admittedly not a mainstream choice, but one that fascinates me. Along these lines, let's say that a person integrates with an irrational as an upper limit, the integral is, I suppose, itself a representative of an equivalence class, but what value does one use for x when giving some non-mathematician an number for applications? –  s.z. Jun 5 '13 at 4:30
    
I like your "primitive notion" concern. Yes, I'm interest in primitive notions. Integers are clear. I don't feel that arithmetic needs a foundation. I even find limits and shrinking intervals clear. I just think the orchestration of these various foundational moves within the context of their application leaves something to be desired. For instance, the exp function is a function of n as well as x, sort of, because we can't actually compute an "infinite" number of terms. So if we integrate exp(x) from 0 to y, with x and y real numbers, we have 3 entangled classes of sequences. –  s.z. Jun 5 '13 at 4:35
    
How far out do we go out on the series expansion of exp? Which term of which representatives of classes x and y do we plug into this series expansion? It's as if we have to boil it all down to rationals at some point, but this boiling-down is not something I see addressed much. –  s.z. Jun 5 '13 at 4:40

The short answer to your question is YES. In mythmatics, the real number is defined as the limit of equivalent Cauchy sequences of rational numbers. The following article is almost the exact mainstream view:

http://www.iitg.ernet.in/kvsrikanth/teaching/2011f/reals.pdf

However, the limit is not always a rational number, in which case, the mainstream idea is that it must be an irrational number. Only problem, is irrational numbers do not exist. Incommensurable magnitudes exist, but not irrational numbers.

A number is the measure of a magnitude.

Since irrational numbers do not exist, it follows that real numbers are a myth.

To get a better idea:

http://johngabrie1.wix.com/newcalculus#!Is-the-real-number-line-real-/c9q/2B33C3CE-9BD0-4872-A290-B57953E94189

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