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.