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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?
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:
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.
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.
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!
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.
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
We understand the rational numbers
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).
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 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.
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.
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.
