Here is my question. Why do the reals need to be "constructed" by this bizarre "Dedekind cut" or "equivalence class of Cauchy sequences" argument? Why can't they simply be "observed" as consisting of all numbers that "span" some known sets of numbers?
I am thinking here, in part, by analogy with linear algebra and with the complex numbers, where $i$, the square root of $-1$, is really all you need, in addition to the reals, to get all complex numbers as a spanning set of $1$ and $i$ over $\mathbb R$. (Every complex number may be expressed as $a\cdot 1 + b\cdot i$ where $a, b\in\mathbb R$.)
We have a couple of known transcendental numbers, $e$ and $\pi$. We have all the rationals. We have all the square roots most of which are irrational. We have all the fractional roots of $e$ and $\pi$. We have the $e$th roots of all the numbers that exist, and the $\pi$th roots. Maybe we also have some other sets of transcendental numbers out there that we can use?
What I am trying to ask is, are we using these "Dedekind cuts" and "equivalence classes of Cauchy sequences" just because we don't "know enough real numbers yet", because their characterization hasn't occurred to us yet, or do we already have enough real numbers in our arsenal, like e and pi, to make a "spanning set" without using equivalence classes of infinite sequences and the like, or, is it the case that we really have to use these kinds of constructions of the reals, for some deep mathematical reason?
It just doesn't seem right. Because you have to admit, by identifying "sets of numbers" like Dedekind cuts and equivalent classes of Cauchy sequences which are both sets of numbers, with actual numbers, mathematicians create (at least in my mind) some cause for doubt about what they are doing here with the reals. A "set" seems like a strangely undefined term, which I understand, but not well, is subject to various kinds of paradoxes and levels of analysis problems. (This last paragraph may be more of a separate question, about the validity of using sets of numbers as numbers, from the first question, which is more about why aren't there simpler ways to define or understand the real numbers in terms of numbers and operations we already understand.)
Best Answer
If you like, and some people do, you can forget about any construction of the reals from the rationals (or anything else) and instead define them axiomatically. One such axiomatization is Tarski's.
This approach will avoid any weird feeling you might have about a real number being an equivalence class of whatnot.
Usually, the reason to provide an explicit construction of something from a simpler things is that it proves that that something exists (mathematically). Moreover, it allows you to study properties of that something in terms of the simpler things that you presumably know better.
Nobody thinks of real numbers as equivalence classes of anything. Once the construction is done you can just forget about it if you like. Having a construction just means that the model of the real numbers that you fantasize about is at least as consistent as a model you might have of the simpler things. To some people it gives reassurance, to others a headache.
As for your attempt to define the read as something spanned by those things we have names for, together with some operations on there. The problem is that there are only countably many such things while there are uncountably many real numbers (at least if you believe that every real numbers admits at most two decimal representations). So this can't work. It might be strange to think about there being more reals then potential names or ways to approximate reals but it's a real fact (pardon the pun).