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My analysis teacher defined the set of reals as, and I quote, a collection of finite numbers $\{A_0, A_1, ..., A_n\}$ and a collection of finite or infinite numbers $\{d_1, d_2, ..., d_m, ...\}$ all in the range $0$ to $9$; we match this collection with

  • $x_+=+A_n...A_1A_0.d_1d_2...d_m$ a positive real
  • $x_-=-A_n...A_1A_0.d_1d_2...d_m$ a negative real

For example, $\pi=3.141...$ is defined such as $A_0=3$, $d_1=1$, $d_2=4$, $d_3=1$, ...

I find this definition very strange, as it seems to be based on syntax alone (or almost). In the rest of the course, we went on to discuss Cauchy sequences and the Cauchy criterion and what it implies:

The sequence $(U_n)_{n\in\mathbb{N}}$ converges if and only if it is a Cauchy one. This equivalence gives $\mathbb{R}$ the property of being a complete set, i.e. any Cauchy sequence of real numbers converges in $\mathbb{R}$.

My intuition then led me to think that the set of reals could be defined as the set containing all the limits of the real Cauchy suites. However, this is only an intuition and I don't know how to show this, I haven't found any result on the internet.

I wonder therefore if this intuition is good, if not (or if it is, because it is always interesting), I would like to know in what other ways we could define the reals, other than by the definition I gave above.

PS: obviously, we could define $\mathbb{R}$ in ensemblistic terms, and say that it contains all naturals, integers, rationals and irrationals, but I was looking for another definition than in terms of sets.

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  • $\begingroup$ Take a look at base 10 representation of numbers and at Cauchy completion of metric spaces. For something a bit different, try Dedekind cuts. $\endgroup$ Mar 14 '21 at 21:32
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    $\begingroup$ Your teacher's definition, as presented here, has $0.999\ldots\neq1$. That's not consistent with standard analysis. More egregiously, it has $0.19999\ldots\neq0.2$, which is dependent on being in a base ten system. Nothing about the real numbers should elevate base ten as a special base. $\endgroup$
    – Arthur
    Mar 14 '21 at 21:33
  • $\begingroup$ Well, you could but then you have to define what a "convergence" means.... okay, that sounds like I'm being critical and faceteous but I'm not. You can define it as the convergences but you have to somehow define convergences without using real numbers. It's equivalent as defining it by bounded sets, or cuts, or cauchy sequences. But... well, you will have to come up with some sort of reference. $\endgroup$
    – fleablood
    Mar 14 '21 at 22:02
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In fact, your teacher's definition doesn't seem to be right per Arthur's comment. I suspect that what they're trying to do is make the construction of the reals as simple as possible, but there is some unavoidable complexity there and - assuming you've copied what they've said correctly - they've stumbled on some of it.

Yes (with a bit of care), this is in fact one of$^1$ the standard ways of constructing the real numbers from the rational numbers.

We start with the notion of a Cauchy sequence of rational numbers only. Intuitively, these are the sequences which should converge but might not ... within $\mathbb{Q}$, anyways! We'd like to add a number corresponding to each such sequence. However, we have to be careful: different Cauchy sequences need not get different numbers! For example, consider $$(3,3.14,3.1415,3.141592,...)\quad\mbox{versus}\quad(3.1, 3.141, 3.14159, 3.1415926,...).$$ These each "point to $\pi$" (ignoring the minor fact that $\pi$ doesn't exist for us just yet!), but are different sequences.

So instead, we work with equivalence classes of Cauchy sequences of rationals. Specifically, there is a natural way - without "looking ahead" to $\mathbb{R}$! - to tell when two Cauchy sequences $(a_i)_{i\in\mathbb{N}},(b_i)_{i\in\mathbb{N}}$ "point to the same thing:"

$\lim_{i\rightarrow\infty}\vert a_i-b_i\vert=0$.

Writing "$\approx$" for the corresponding equivalence relation, we define $\mathbb{R}$ to be the set of $\approx$-classes of Cauchy sequences. Addition, multiplication, and so on of such classes can then be straightforwardly, if somewhat tediously, defined.


$^1$There are actually many different ways to "construct the reals." My personal favorite is via Dedekind cuts, FWIW. Given the plethora of options, this raises an interesting methodological concern: what exactly do we mean by "construct the real numbers," or even "the real numbers" for that matter? If I use a different construction than you, do we "disagree about $\mathbb{R}$?"

Addressing this concern satisfactorily turns out to be rather involved. I'm mentioning it here, however, since I think it is one which can reasonably occur early on and will only cause confusion if swept under the rug. So, even though it uses jargon which presumably you have not seen yet, I think it's worth stating the key theorem if only so that you know that such a thing exists:

There is exactly one complete real closed field up to isomorphism.

While the precise meaning of the above probably isn't clear, the general idea is quite simple: even if you and I think of $\mathbb{R}$ in terms of different constructions (e.g. Cauchy sequences vs. Dedekind cuts), "your version of $\mathbb{R}$" and "my version of $\mathbb{R}$" will be basically identical since they'll share a couple key mathematical properties. It turns out that there is a lot of subtlety here - see e.g. the discussion here - but ultimately the point is sound.

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You have very good instincts. There are a couple of points to note. First, your teacher's definition is not in fact well-defined, because two different sequences may end up defining the same number; e.g., $1=0.999 \ldots$.

Second, it is indeed possible to use Cauchy sequences to define the reals. In fact, it's a standard method of doing so. However, it's a little more intricate than you may realize.

First you define the natural numbers. Then you use natural numbers to define the integers -- an integer is an equivalence class of ordered pairs of natural numbers under the equivalence relation $\forall a, b, c, d \in \Bbb N~((a, b) \sim (c, d) \iff a+d = b+c)$ (Think of the ordered pair $(a, b)$ as $a-b$.)

Then you use the integers to define the rational numbers: $\forall, x, z \in \Bbb Z, y, w \in \Bbb Z \setminus \{ 0 \}~((x, y) \approx (z, w) \iff xw=yz)$. You can think of the ordered pair $(x, y)$ as $\frac xy$. However, you need to confirm this proposed definition is in fact well defined: In other words, if $x_1, x_2$ are different ordered pairs of natural numbers such that $x_1 \sim x_2$, and $y_1, y_2$ are different ordered pairs of natural numbers such that $y_1 \sim y_2$, then you need to confirm $(x_1, y_1) \approx (x_2, y_2)$.

Once you have the rational numbers, you can define Cauchy sequences of rational numbers. There are lots of different Cauchy sequences that converge to the same number, so you need to define an equivalence relation among Cauchy sequences. One that works is that two Cauchy sequences are equivalent if the "interleaved" sequence also is Cauchy.

The set of equivalence classes under this equivalence relation is the set of real numbers.

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Your teacher seems be taking the "naive" and preliminary school definition of real numbers as any number can be written as decimals but the decimals might need to go on forever. But then wants to make it formal and rigorous.

It seems to me your teacher thinks this is the best of both worlds but IMO it is the worst of both worlds.

It does work. We can get to any value with a $\frac 1{10^k}$ accuracy by taking an number with a $k$ decimal place and we can hone in to finer degree by taking a $k+1$ decimal places. So if we take an infinite sequence $a_k$ where $a_k$ has $k$ decimal places and $a_{k+1}$ has $k+1$ decimal places and $|a_{k+1} - a_k| < \frac 1{10^k}$ that is all the $k$ decimal places of $a_{k+1}$ are the same as $a_k$'s decimal places. then $a_k$ is a cauchy sequence and $a_k \to a_\infty$ an decimal with an infinite number of decimals defined to be $\lim\limits_{k\to \infty}a_k$.

(In other words if $a_0 = 3; a_1 = 3.1; a_2 = 3.12; a_3=3.141; a_4 = 3.1415$ and $a_k=\pi$ rounded down to $k$ decimals, the $\{a_k\}$ is a cauchy sequence and $\pi = \lim a_k$.)

The subtle and difficult thing about define the reals is you don't have a "place" in which to define them. The intuitive reason for the decimal definition as that it is taken for granted that numbers "live" in a continuous infinitely precise space and since decimals can hone into everywhere it must be that every point in in the continuous space can be honed into be an infinite decimal expansion.

That sort of begs the question though.

Grown up mathematicians make an abstract conception. The no that the rationals aren't good enough-- they aren't complete as the "jump" from below an irrational such as $\sqrt 2$ or $\pi$ to above without ever being able to hit $\sqrt 2$ or $\pi$. And they can't say $\lim a_k =\pi$ because that $\pi$ doesn't "live" anywhere? Where does $\pi$ live?

They say. If we can invent a space that isn't the numbers but behaves exactly like the numbers then it is equivalent and for all intents and purposes is.

And they do exactlY what you suggest.

The take the set of all cauchy sequences. And the say two cauchy sequences are essentially the same as each other if their terms eventually always get arbitrarily close to each other. And thus they define the real numbers as basically being identified by a cauche sequence.

So we'd say $\pi$ is essentially the same thing as $\{3,3.1,3.14,3.141, 3.1415.....\}$ and that is essentially the same thing as $\{4, 3.2, 3.14, 3.142,3.1416,.....$.

And yes, that does work. But there are a lot if details I glossed over.

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You are quite close. The problem is that before you define real numbers, a Cauchy sequence that ought to converge to the squar root of 2, for example, has nothing yet to converge to.

However, even without having real numbers, you can define whether two Cauchy sequences converge to the same number, and that’s when the differences between the elements of two Cauchy sequences get closer and closer. So you can split the Cauchy sequences into equivalent classes of Cauchy sequences that converge to the same limit. And you can use these equivalence classes to define the real numbers.

(That’s one of the two most common methods to define real numbers actually, the other is to define a real number as a non empty set of rational numbers with an upper bound that is not an element of the set, with the property that if x is an element of the set and y < x then x is an element of the set as well.

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