Can we define the set of reals as the set containing all the convergences of the sequences

elementary-set-theoryreal numbersreal-analysis

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.

Best Answer

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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