Why is the construction of the real numbers important?

There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers as a complete ordered field?

What's the importance of studying the construction of the Real numbers? Is it just for historical reasons?

• Why does there exist a complete ordered field? – LASV Apr 1 '15 at 18:07
• When you set down a system of axioms for a theory, it is good to know that there is a model for that theory, in other words a set with the proper relations that satisfies those axioms. By rigorously constructing the reals we show that there is a complete ordered field, at the same time convincing ourselves that the real numbers have a valid basis. – Joe Johnson 126 Apr 1 '15 at 18:12
• “The method of postulating what we want has many advantages; they are the same as the advantages of theft over honest toil.” -Bertrand Russell, Introduction to Mathematical Philosophy – Achilles Apr 1 '15 at 18:31
• While we're at it, why don't we simply define the Natural Numbers as a structure satisfying Peano's Axioms, Fermat's Last Theorem, and Goldbach's Conjecture? Think of all the work that could have been saved. – WillO Apr 2 '15 at 13:03
• Consider a course that starts with "Let $F$ be any complete ordered field ..."; then shows a large amount of properties of such fields; then shows that any two such fields are isomorphic; then shows that the only field constructed so far ($\mathbb Q$) is not an example; then shows that one can construct an example a la Cauchy or Dedekind. I think such a course might be quite amusing, both for the "surprise" at the end and for the fun fact that no "dirty" Cauchy sequences or Dedekind cuts were needed for the first part. – Hagen von Eitzen Apr 2 '15 at 19:07

First of all, mathematics is based on intuition and on concrete (imaginary, but concrete) objects, often inspired by reality. Let's stop the formalities for a moment and speak freely: we don't think of $3/4+1/2$ as an operation involving equivalence classes of ordered pairs of Von Neumann integers, we think of it as pouring $3/4$ liters of water and $1/2$ liters of water into a bowl.

With that said, if you're going to be doing reasoning on the basis of intuitive objects like fractions and integers, you need somewhere to start. I'm not even really talking about axioms, I just mean that you need to accept that certain things are reasonable enough to just accept that you understand how they work without having to analyze any further. I accept whole numbers as a basic object of mathematics, I don't need to ask what a whole number is or what it means to add whole numbers. I suppose all I'm saying is that in mathematics, we need undefined objects.

Now, I'm sure you're willing to accept integers, integer addition and whatnot as undefined objects. Probably you accept rational numbers as well - you know what I mean when I talk about "chopping $3$ things into $4$ equal pieces", and what it would mean to "add" two such quantities.

So, so far we're agreed that accepting rational numbers as basic objects without further analysis is philosophically tolerable. Well, I'm sure you've seen the proof that the square root of $2$ is irrational. But check it again - that's not what it proves. What it proves is that there is no rational number which squares to $2$. It doesn't prove that there is such a thing as irrational numbers or that there's any object in the universe worth calling a square root of $2$. In fact, if so far we've accepted rational numbers into our menagerie of philosophically coherent objects, then there's no reason at all why we shouldn't simply stop here and say "well, clearly there's no square root of $2$". Let's be honest, the reason why most students feel so strongly otherwise is because they've never gotten ERR when they typed SQRT(2) into a calculator - it's accepted on the basis that an authority figure told them so, and the philosophy of it all goes unquestioned. But there really is no reason to panic and postulate a bigger number system just because there's no number whose square is $2$, there's also no number whose square is negative, but that didn't bother anyone until they started doing relatively advanced algebra.

But hold on, there is a way to rescue poor $\sqrt 2$. Let's say that you needed a number whose square was $2$ - maybe you needed to draw a square whose area was $2$. Well then you wouldn't be bothered by my ridiculous metaphysical musings, you'd just pick a rational number whose square was close to $2$, say $1$cm and $4$mm, and draw the square that way. Now, if we want better and better approximations - number whose squares are closer an closer to $2$ - we would find that these numbers "converge" to a sort of "ideal point". For example, given any $\epsilon$, I can find an interval of width $\epsilon$ in which all approximations beyond a certain precision must lie. What's more, these intervals will nest in one another as $\epsilon$ decreases, so they really are "tightening" around a specified point.

It's tempting to call this "point" a number, but that's unacceptably vague for a mathematician. What is this "point", for one thing? Certainly not a rational number (the only kind of number we understand, so far). One approach is to say that the only reason we believe this "point" exists is because we have a sort of oracle that tells us, given any rational, whether it's "too big" (it's square is bigger than $2$) or "too small". It's by the use of this oracle that we can find better and better approximate square roots of $2$: make a guess, then make it a little bigger or a little smaller depending on if its square is smaller or bigger than $2$. Thus we might say that any time we have such an "oracle", we can claim to have found one of these mysterious idealized "numbers", which can be approached but never precisely given a value. This is essentially a Dedekind cut.

Hopefully I've convinced you of two things:

1. There is no obvious reason (other than constant mindless drilling in mandatory education with symbols like "$\sqrt\cdot$" and "$\dots$") to suspect anything resembling the real numbers exists or is worth discussing.

2. Nevertheless, with some reflection such a reason can be found (otherwise the real numbers would never have been developed, of course!), and it leads very naturally to the various constructions of the reals.

Now, if you accept (1), then the answer to your question is simple. If it's not obvious that the reals exist, then postulating their existence is absurd. Even if you're okay with the idea of just writing down and investigating some random axioms, there would be no reason at all to suspect these axioms describe the real world, or anything interesting at all, in any useful way. That last point is crucial. The constructions of the real numbers are the only reason to think that real numbers bear any relation to reality at all. Proving that a complete ordered field exists in ZFC, as pointed out in the other answers, is neat, but it's not really the most important reason to construct the real numbers, since merely existing doesn't imply that a structure is interesting or is a valid model for real-world quantity.

And consider (2). As we saw above (in a very summarized way), by the time you've given the question enough thought to convince yourself that there's any such thing as a real number, you're half way to rigorously constructing the real numbers anyway, so you may as well finish the job.

• To draw a square of area two, first draw a square of area one and build a square on the diagonal... (See Meno by Plato.) – Daniel Juteau Apr 1 '15 at 20:10
• Nice answer. You should patent your idea of a Rational Calculator. – Keith Apr 2 '15 at 2:40
• I disagree with your notion about rationals being atomic, rather than equivalence classes (and the appeal the von Neumann "integers", those are just the natural numbers, and the name dropping in order to sound more ridiculously formal is also meh). Do you think about fifteen minutes as $1/4$ of an hour? How is this not using the equivalence between $15/60$ and $1/4$? – Asaf Karagila Apr 2 '15 at 7:45
• @AsafKaragila If rationals are just equivalence classes, how do you justify their relation to the real world? How do you justify that chopping something into three pieces gives $1/3$ of what you originally have? Also, you then have to simply define the operations somewhat arbitrarily, when deep down you secretly know that those operations can be proven (informally, but still convincingly) to be correct. This dissonance between the formalization and your intuition means that the formalization isn't really useful, and rationals are pretty much just fundamental objects. – Jack M Apr 2 '15 at 13:38
• Jack, if I gave you five rocks, and then took away three rocks. You are not left with $2/5$ of the five rocks. You are left with two rocks. And what if I gave you ten rocks and took six? Will you be left with $2/5$ of the ten rocks, or will you have left $4/10$ of the ten rocks? Again, you'll be left with four rocks. The ability we developed to think in fractions doesn't mean that rational numbers are "deeply connected" to the world. It just means that the world is made of smaller bits that we can group together and think of as one object for a moment. – Asaf Karagila Apr 2 '15 at 13:53

Imagine a mathematical paper that starts with: "Let $\mathbb G$ be a countable complete ordered field." We know that such an object doesn't exist, so anything that the paper has to say about it is vacuous.

Now imagine a paper that starts with: "Let $\mathbb R$ be a complete ordered field." Thanks to our constructions, we know that such an object does exist. This is a crucial difference!

• Although I agree with this philosophy, it leaves me deeply perturbed when I read a paper starting "Let $\kappa$ be a measurable cardinal...". – Mario Carneiro Apr 4 '15 at 3:36
• @MarioCarneiro: Are you deeply perturbed when you read a paper that uses ZFC for its metamathematics? At some point, unfortunately, we have to take things for granted or else treat everything as hypothetical. Or restrict ourselves to ultrafinitism or something, but where's the fun in that? – tomasz Apr 5 '15 at 8:12
• @tomasz My point is that most large cardinal theory actually takes place in ZFC, with actual existence axioms being used sparingly. This makes most of the theorems take on this strange philosophical tone since they prove properties about an object that is not known to exist (and indeed cannot be proven to exist). This differs from regular infinity in that we generally take this as an axiom, so that one can actually say that the objects exist in the theory, rather than just proving conditional statements like "if $\kappa$ is inaccessible then $V_\kappa$ is a model of ZFC" (in ZFC). – Mario Carneiro Apr 5 '15 at 19:22
• @MarioCarneiro: Thank you for your comments! But there is a big difference between "not known to exist" and "known not to exist". – TonyK Apr 5 '15 at 20:06

You need to motivate the readers, at least newcomers to mathematics. If I were to tell you that there exists an object such and such, why should you believe its existence is plausible? How can you tell that its existence is not contradictory?

You won't, or at least shouldn't. And it's fine.

But, if we sort of agree that the natural numbers exist and by proxy the rational numbers exist, then we can construct the real numbers, and thus prove that they exist.

This is also important because it is a good way to introduce the notions of sequences, convergence, linear order, and so on and so forth, and accustom the readers to the fact that in mathematics we try not to take too many things "on faith", but rather begin with a very small set of assumptions and build the rest of the universe.

• +1 for "in mathematics we try not to take too many things on faith". This is an attitude which one should have when studying mathematics. – Paramanand Singh Jun 9 '16 at 12:14
• @ParamanandSingh "Too many things" refers to quantity, however, not to be picky or anything, I think it's about the quality of formulae that should decide whether or not we take it on faith. For example, if I have twenty or so conjectures, supported each with abundant numerical evidence, it would be reasonably fair to assume them valid. The "line" that mathematicians cross when "taking a formula on faith" is not fixed, and is different for everyone relevant to their intuition and subjective taste of "quality". Ramanujan, putting aside his unorthodox methods, is an excellent example. – Mr Pie Apr 30 at 12:51
• @MrPie: the case of Ramanujan is no different in this regard. Ramanujan obtained many of his result by empirical evidence. Contrary to popular belief he proved all his results but due to lack of paper he did not record most proofs. For results where he did not have proofs he would write explicitly that there was no proof. Rogers Ramanujan identities were empirically found. Later Ramanujan saw a proof by Rogers while in England and was surprised. Later he developed his own proof different from that of Rogers. – Paramanand Singh Apr 30 at 13:24

In traditional geometry, the dispute over the parallel postulate was whether the parallel postulate was provable from the other axioms. If it was, then there would be no need to include it as a separate assumption. This is a general aspect of mathematicians: we like to have fewer, rather than more, assumptions, whenever possible.

It turns out that the standard mathematics that is usually taught in an undergraduate mathematics degree - calculus, basic real analysis, basic abstract algebra, etc. - can all be derived from a particular set of axioms, which are axioms for set theory.

The axioms of set theory include an axiom - the axiom of infinity - that allows us to directly construct the natural numbers. From the naturals, we can construct the integers and rationals in a direct way.

The construction of the reals from the rationals, carried out in set theory, shows that no new axioms are needed for the real line. So, unlike the parallel postulate that must be assumed in addition to Euclid's other postulates, we do not need to assume there is a complete ordered field: we can construct one using the axioms we already have.

The construction also shows that if

1. We understand the rational numbers

2. And we understand subsets of the rationals well enough to understand the "completeness" axiom

then that alone is enough for us to understand the real line. In other words, it is completeness alone - and not really any other property of the reals - that leads to their uncountability.

Finally, the construction of the reals using Cauchy sequences is also very relevant to computation. It is completely possible to write programs that compute with real numbers, by representing the real numbers as Cauchy sequences of rationals. This is called "exact real arithmetic" (this is not the same as "arbitrary precision arithmetic" which is really only about rational numbers).

• I think for honesty you should add that in exact real arithmetic, there is no (total) comparison operation like $\leq$ or $=$. This makes algorithms for computation rather different than those studied in numerical analysis. – Marc van Leeuwen Apr 4 '15 at 13:06

For one thing, as I stated in my comment, there is the question of existence. Is it clear to a novice that there exists such a complete ordered field? For another thing, what about uniqueness? Are the real numbers completely determined by these axioms? Or could there be a nonisomorphic field that is complete and ordered?

• We can use the axioms of the complete ordered sets to prove there is just such one, we don't need the construction of the Real numbers to prove it. Am I right? – user42912 Apr 1 '15 at 18:26
• You are correct. You can prove it. But I am just trying to make a case that when you define things by just axioms, there are sticky points to clear up. In this case, existence is the main one. Another interesting thing is that using the idea of the construction of the real numbers as the completion with respect to the euclidean metric, one can build other highly interesting fields, such as $\mathbb{Q}_p$, the $p$-adics. – LASV Apr 1 '15 at 18:28
• I think it is clear enough to a novice that $\mathbb{R}$ is a complete ordered field - which is exactly why it's worth asking why we need to construct it! – Carl Mummert Apr 2 '15 at 0:38

We often do define the real numbers as a complete, ordered field.

A construction $\mathbb{R}$ from $\mathbb{Q}$ is a demonstration of technique.

For example, a student of real analysis is likely to learn study metric spaces. They will learn how to consider functions on a dense subset and how their study relates to functions on the metric space. They will learn to construct the Cauchy completion so as to view the metric space as a dense subset of a complete metric space.

Introducing these concepts with a simple, familiar example is a useful demonstration of how to use these ideas.

Similarly, the proof that the construction constructs a complete ordered field will introduce new ideas and demonstrate how to use those ideas to prove things, thus helping a student start to think in terms of these new techniques.

People have worried about Real Numbers for a very long time (see Stephen Harking: God Made the Integers). When I did maths at university, I was interested in mathematical physics, and I initially accepted the facile answer that we need real numbers for physics, so they must exist. But what if reality is quantized? What if all lengths are integer multiples of the Planck length? I wouldn't want to throw out calculus if it turned out there were no real (irrational) numbers in physics. Constructions such as the Dedekind Cut mean that we can have real numbers and calculus regardless of what "reality" chooses to do.

• If all lengths are integral multiples of the Planck length, what do we do about Pythagoras' theorem? – Marc van Leeuwen Apr 3 '15 at 13:30
• @MarcvanLeeuwen Take the closest integer multiple of the planck length. – user142198 Apr 4 '15 at 3:44
• That is, you assert that space does not have the "expected" continuous metric and invent some rules for what metric it does have. The actual "distance" might be the closest integer multiple, or in some cases it might be another nearby value ("nearby" we know because the continuous metric appears to be a good approximation so far). You'd have to come up with a theory that says what, and test it :-) Anyway perfect "triangles" in the usual sense needn't exist. What exists is some ape's attempt to make one out of bits of tree or whatever. – Steve Jessop Apr 4 '15 at 12:44

There is a calculus textbook, "Mathematical Analysis I" by V. Zorich, that starts with $\mathbb R$ and its properties taken for granted. It then derives $\mathbb N$ and $\mathbb Q$ from $\mathbb R$, and proceeds to actual calculus.

Usual construction of $\mathbb R$ from $\mathbb Q$ is given as a supplemental material, if I recall correctly. So, in a sense, such a construction is not actually crucially important.

• I couldn't take any text that derives $\Bbb N$ from $\Bbb R$ rather than the other way around seriously. Concretely this approach suffers from the fact that the definition "complete" refers to Cauchy sequences, which are indexed by (something like) natural numbers. That is a bit problematic if you haven't got the natural numbers yet. I suppose this specific difficulty can be circumvented with different definitions (after all one can introduce natural numbers using nothing but set theory), but it still remains a fundamentally backwards approach. – Marc van Leeuwen Apr 4 '15 at 12:57
• @MarcvanLeeuwen That's what I thought too when I first heard about this approach, but maybe it's not. For the completeness, the Dedekind's definition is used, and the author really sets to step away from the classical, XIXth century approcah to calculus to more XXth century one: so you see limits on filters, differential as a linear mapping, integrating differential forms on varieties—and all this is in a calculus course for the first two years. – Joker_vD Apr 4 '15 at 15:58

The greatest tragedy with mathematicians is that we can't talk about anything whose existence has not been proved a prior. Further, typically three fourth of all mathematical ingenuity is used up in proving "existential theorems". In real analysis, assuming the existence of rational number system, we exhaust much energy in proving that there exists a complete ordered field. The members of the complete ordered field are called real numbers. Using this definition of real number the whole edifice of real analysis is developed. Philosophically, this is the correct way of learning real analysis. On the contrary, suppose that complete ordered field would not have existed. In that case, many statements would have appeared which are true and false both. In mathematics, we deliberately avoid such circumstances.