I recently read that a real number $r$ is any subset of the set $\mathbb{Q}$ of rational numbers such that it satisfies the following.

1) $r$ is nonempty

2) $r\neq\mathbb{Q}$

3) $r$ is closed downwards

4) $r$ contains no greatest element

My question is how can a single real number $r$ be a set of rational numbers?

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    $\begingroup$ The conceptual obstacle here, that how a single real number can be a set of rational numbers, is answered much more clearly by something you already know: decimal expansions. Any real number is truly just the set of rational numbers given by cutting off it's decimal expansion at the $n$-stage, for $n = 1, 2, \cdots$. The idea here is similar, just the process is different (This has a name, by the way. It's called the Cauchy completion - which is another way of construction of the real numbers). $\endgroup$ – Balarka Sen Jan 9 '18 at 4:18
  • $\begingroup$ Well for that matter all the numbers (naturals, integers, rationals, complex ones) are some kind of sets. Why do we find the idea of a real number as a set difficult to believe? $\endgroup$ – Paramanand Singh Jan 9 '18 at 5:48
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    $\begingroup$ It's not really different from a rational number being a pair of numerator or denominator (actually, an equivalence class of such pairs, because $1/2=2/4=3/6=\ldots$). You might just as well ask "how can a number be a pair, or set of pairs?" $\endgroup$ – celtschk Jan 9 '18 at 7:24
  • $\begingroup$ Side note : the thing that has made me understand Dedekind's cut is when someone told me to think of it as a clever way to ensure that $\mathbb{R}=\bar{\mathbb{Q}}$ $\endgroup$ – Evpok Jan 9 '18 at 14:47

To elaborate perhaps a bit on what Y. Forman wrote, it's more that we think of the real numbers as the Dedekind-complete ordered field - they satisfy the field axioms (just like $\mathbb{Q}$, for example) and they have the least-upper-bound property (that any set $S \subset \mathbb{R}$ such that $\forall s \in S, s \le b$ for some $b \in \mathbb{R}$ means there is a least $\lambda \in \mathbb{R}$ such that $\forall s \in S, s \le \lambda$).

What the Dedekind cuts are is a specific construction of a Dedekind-complete ordered field. There are actually different constructions! But, they are all the same in an important sense: they are isomorphic.

Proving this stuff is an interesting exercise - but the takeaway is, you don't need to think of the reals as being Dedekind cuts. Rather, we're proving that the concept of Dedekind-complete ordered field makes sense, by showing that it can be constructed - e.g., as Dedekind cuts - and then regardless of how it's constructed, we use the properties of "Dedekind-complete and an ordered field" to deduce results.

  • $\begingroup$ Well whatever be the construction of real numbers it is important to understand and a real number is described by a set of rational approximations and that set has to be infinite. Often when people try to picture a real number as a point on some number line, this infinite stuff gets hidden from view. $\endgroup$ – Paramanand Singh Jan 9 '18 at 5:54

The purpose of Dedekind cuts is to define the real numbers, given that all we know is the rational numbers. The intuitive grasp of the concept of real number is not enough. If we define a real number as a set of rational numbers, then a real number is a set of rational numbers.

Since this definition doesn't match your intuitive grasp of real numbers, the definition of concepts like addition of real numbers also won't match your intuitive grasp.

  • $\begingroup$ Also what's the intuitive grasp of a real number? Everyone seems to have some intuitive idea of real number, but the intuitive idea is never pointed out clearly. $\endgroup$ – Paramanand Singh Jan 9 '18 at 5:56

The idea of a real number is somewhat difficult to grasp, but it is very important to have that idea if one is serious enough to study advanced mathematics. A typical encounter with real numbers in high school starts with the number $\sqrt{2}$ and one is told that this $\sqrt{2}$ is not a rational number. Most common textbooks don't emphasize the fact that the actual result proved is that there is no rational number whose square is $2$ rather than there exists some number of a different kind whose square is $2$.

One can fix this by postulating the existence of such numbers and their cousins like $\sqrt[3]{5}$, but using such algebraic procedures based on roots of certain equations does not give us the true picture of real numbers. In fact any approach based on using a finite set of rationals to describe a real number fails miserably.

Reals numbers are essentially a set of rational approximations (to what??) and the key property here is that one can find as good approximations as needed. And this necessarily means that the set in question has to be infinite. For if the set were finite one of the members would turn out to be the best approximation and we won't have a better approximation than that member.

Formal definitions of a real number use specific kinds of infinite sets of rationals to describe a real number and thus different approaches may have different sets being used to describe the same real number. For example we have Dedekind cuts (given in your question) or Cauchy sequences (and what not). The situation is similar to the fact that a complex number can be described as a pair of real numbers or as a coset in the quotient $\mathbb{R} [x] /(x^2+1)$. Individual constructions of complex numbers don't matter but what matters is that there are two unique real numbers involved in describing a complex number.

The situation for a real number is complicated on two fronts: first of all an infinite set of rationals is involved and moreover there is no such unique set involved in defining a real number. But one should not be afraid of such a scenario and instead think of decimal representation. The rational number $1/3 $ has a non-terminating decimal representation. We essentially mean that the real number $1/3$ can be represented as the infinite set $$\{0.3,0.33,0.333,\dots\} $$ and in a similar manner the real number $\sqrt{2}$ is represented by the set $$\{1.4,1.41,1.414,\dots\} $$ Also note that in all these representations the important elements are not those in the beginning, but rather the ones hidden behind ellipsis $(...) $. One might prefer to define the symbol $\sqrt{2}$ as the set of values taken by sequence $x_n$ defined recursively as $x_1=1,x_{n+1}=(x_n+(2/x_n))/2$ or go the Dedekind way and use the set $$\{x\mid x\in\mathbb{Q}, x\geq 0,x^2<2\}\cup\{x\mid x\in\mathbb{Q}, x<0\} $$ It takes some patience to understand in what sense these representations describe the same real number $\sqrt{2}$ and what is common between different representations of a real number.

The idea of presenting real numbers as axiomatic objects without any underlying structure simply misses the point. And perhaps it is a very step fatherly approach towards real numbers. After all other kinds of numbers have a concrete representation and that forms the key to their properties why this kind of inferior treatment to real numbers? I don't know how the axiomatic approach became a hit among educators, but IMHO real numbers are much more concrete than other kinds of numbers. One should try to get used to their representation via infinite set of rationals and move on to study their key property of completeness.

  • $\begingroup$ In whiich sense are real numbers much more concrete than other numbers? I'd say the natural numbers are at least as concrete as the real numbers. $\endgroup$ – celtschk Jan 9 '18 at 7:41
  • $\begingroup$ @celtschk: they have much more beautiful structure. In fact it is so beautiful and deep that it needs an infinite number of rationals to describe it. $\endgroup$ – Paramanand Singh Jan 9 '18 at 8:10
  • $\begingroup$ Deepness and concreteness are two unrelated concepts. And how can something be more concrete than whatever is needed to describe it? $\endgroup$ – celtschk Jan 9 '18 at 8:18
  • $\begingroup$ @celtschk: I am not saying that a thing is more concrete than the thing used to describe it. Thus a real number is not any more concrete than the infinite set of rationals used to describe it. But an individual real number is more concrete than an individual rational number. By concrete I mean having some underlying structure. $\endgroup$ – Paramanand Singh Jan 9 '18 at 9:32
  • $\begingroup$ “By concrete I mean having some underlying structure.” That is not the meaning of “concrete”. No wonder I didn't understand you. $\endgroup$ – celtschk Jan 9 '18 at 21:20

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