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I want an intuition of how this set is constructed more than a formal proof. At first I thought that the set was simply defined axiomatically but further reading showed me that there had been attempts to construct the set explicitly like the construction from Cauchy sequences. So what are the real numbers ? An axiomatically defined set? the completion of the rational numbers? Both? Something else?. Thanks a lot in advance!

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In mathematics, we often don't really care what something "is" in some fundamental sense, but what its properties are. In this way, we may view the real numbers as any complete, ordered field $\mathbb{R}$ which contains the rational numbers as an ordered subfield.

At this point, you may have two questions: does any set of real numbers $\mathbb{R}$ exist and, if so, are there more than one which are "different" from each others. As you alluded to, there exist several constructions, most famously in terms of Dedekind cuts and Cauchy sequences of rational numbers, which show the existence of a set of real numbers. It is also true that the real numbers are unique. That is, if I have two complete, ordered fields $\mathbb{R}, \mathbb{F}$ containing the rationals as an ordered subfield, there there exists an order-preserving isomorphism between them. In other words, $\mathbb{R}$ and $\mathbb{F}$ are the "same".

Note that the Cauchy sequence construction shows that the real numbers are the metric completion of $\mathbb{Q}$ under the metric $d(x,y) = |x-y|$, so we may additionally view $\mathbb{R}$ as the metric completion of $\mathbb{Q}$.

What is worth noting is that, for any practical matters, it doesn't matter what model of the real numbers you use. Whether real numbers are equivalence classes of Cauchy sequences, Dedekind cuts, or something else doesn't effect the actual properties of the real numbers that you care about for analysis.

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  • $\begingroup$ Small caveat: the reals cannot actually be constructed as a metric-completion of the rationals, because the standard definition of metric-spaces rely on the reals. However, if you relax the definition and only require the metric to have values in some ordered semi-group, then we can indeed prove that the rationals form a generalized metric-space and we can construct its metric-completion via equivalence classes of Cauchy sequences. =) $\endgroup$ – user21820 Apr 24 '17 at 12:02
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There are many answers to this question on the site but I'll give a fairly intuitive (to me) one here, starting from natural numbers (go to paragraph 4 for just rationals to reals):

Start with natural numbers, that is, whole numbers or counting numbers starting from 0. For every natural number $n$ that isn't zero, define $-n$ to be a new number so that $n + (-n) = 0$. The set of all these numbers together makes the integers. Note that if we add or multiply any two numbers in this set, we get a third number that's also already in the set.

Then, for any integer $z$ that isn't zero, define $z^{-1}$ to be a new number so that $z\cdot z^{-1} = 1$. If you add or multiply any two numbers in this set, you might get a new number not in the set. For any integer $z$ and natural number $n \neq 0$, define a new number $z\cdot n^{-1}$. The set of these numbers is called the rational numbers, and any sum or product of these numbers is another rational number.

Now, you might be tempted to believe that this is all the numbers there are. In between any two rational numbers, you can find an infinite number of rational numbers, and you can pick two rational numbers to be as close as you want. However, you want a number system so that you can easily find a number $r$ so that $r^2 = 2$ (or $r^n = m$ for natural numbers $m,n$), but it can be shown that there is no rational number that does it. In fact, we can show that a number that solves this equation will have a non-repeating decimal expansion, while all the rational numbers repeat (with terminating decimals ending in repeating $0$'s). So how can we get all the numbers with infinite decimal expansions?

Start with a non-repeating sequence of natural numbers from $0$ to $9$, called $d_n$. Then, define a new sequence $a_n = \sum_{i=0}^n d_i\times10^{-n} = d_0.d_1d_2d_3 \dots d_n$. Then, as $n$ gets bigger, $a_n$ gets closer and closer to a particular number $a$. While each $a_n$ is rational, $a$ is not rational since it requires an infinite number digits, so we define $a$ to be a new number. The set of numbers we have now is the real numbers, and can be shown to follow all the "axioms" of real numbers you've learned. Every continuous function $f$ which has positive and negative values for some real numbers has a solution to $f(a)=0$, so most of our "intuitively solvable" kinds of equations can be solved with these numbers.

However, these numbers don't do everything, as something like $a^2 = -1$ doesn't have a solution ($f(a) = a^2+1 > a^2 \geq 0$, so $f(a) \neq 0$ for any real $a$). We use this to define complex numbers, which themselves can't solve every equation, such as $(ix-xi)^2+1=0$.

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The real numbers $\mathbb{R}$ is the unique complete order field up to isomorphism-- $\mathbb{R}$ has cardinality $2^{\mid \mathbb{N} \mid}$.

You can construct the natural numbers $\mathbb{N}$ recursively. From which we can construct $\mathbb{Z}$: the integers, next we construct $\mathbb{Q}$: the rationals and finally from $\mathbb{Q}$ we construct $\mathbb{R}$: the real numbers via Cauchy sequences, Dedekind cuts, or another method.

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There are multiple ways to construct the real numbers. As you have mentioned, they can be viewed as a complete order field. Complete here meaning that every nonempty set has a least and greatest upper bound.

There is my favorite way which is to view $\mathbb{R}$ as the completion of $\mathbb{Q}$ with respect to the metric |x-y|. Completion here is that all Cauchy sequences are convergent where the limit lives in your space. This procedure is nice since you can generalize this to metric spaces, namely every metric space has a completion. Depending where you are at you should prove this! There are books that walk you through this in the exercises. Royden is one of them.

You can also construct the reals by Dedekind cuts. I personally do not know much about this but I am sure a quick google search will produce many resources.

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The simplest and most intuitive (and involving least technical machinery) approach to construction of real numbers is the one given by Dedekind in his famous pamphlet Stetigkeit und irrational zahlen in German. An English translation by the name of Continuity and irrational numbers is a very good read. Unfortunately modern textbooks of analysis present this construction in a very boring manner full of symbolism/formalism and you should seriously avoid such presentation/treatment.

Further it has been the trend of modern textbooks to disregard the importance of construction of real numbers so much so that most textbooks would rather convince you that it is a totally pointless exercise.

It is not really important to study a formal proof of construction of real numbers (because a large part of it is very very boring and it is a total waste of time). But there are two things which every student of analysis must understand:

1) The relation of real numbers to rationals. Just like we can define rationals as ratio of two integers and get all properties of rationals from this simple fact, it should be possible to define (and make sense of) real numbers in terms of rational numbers.

2) One must be able to know how real numbers satisfy the completeness property based on whatever way we define real numbers in terms of rational numbers. Axiomatizing this part is more akin to memorizing theorems without proofs (and learning to apply the theorems without their proofs).

In other words we should not be worried so much to prove that real numbers form an ordered field, but we must be worried to know how they form a complete ordered field. The boring parts of the construction of real numbers are related to proving that real numbers form an ordered field. The part about completeness is never boring if presented in right fashion (i.e. first showing the inadequacy of rationals in terms of completeness and then showing how to fix this problem) and it is the key to all the significant theorems in elementary analysis/calculus.

Apart from Dedekind's original pamphlet the only good (meaning not boring) treatment of the construction of real numbers which I have found is in Chapter $1$ of G. H. Hardy's classic A Course of Pure Mathematics.

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