Negation of the Completeness Axiom There are a lot of theoretical results in Real Analysis, Calculus, Topology, etc. that depend on or use this result below. This has been a question that has bothered me for a long time and seems to me very similar to Euclid's Fifth Postulate in Geometry. Negating the fifth postulate has led to different types of geometry which is why I am curious about the theoretical nature of how real numbers behave when negating the least upper bound property (i.e. given below).
The Completeness Axiom states every non-empty set $S$ of real numbers that has an upper bound also has a least upper bound. Sometimes this is a theorem that relies on other axioms which I find also terribly non-intuitive. Like why does this property have to work? It doesn't seem like a typical evident axiom like so many other theoretical objects are in math.
What I was wondering is what happens when we negate the completeness axiom? In other words, there is some non-empty set $S$ of real numbers that has an upper bound but lacks a least upper bound. I am curious as to what happens. So, my question is what happens when we negate the completeness axiom?
 A: By the real field axioms, you have the $0$, the $1$, $1>0$,  and because you need all the inverse elements for addition and multiplication (except for the 0), you have at least the $(\mathbb{Q},+,\cdot)$ field.
A possible characterization of the sets you are interested may be the sum of rational numbers with all rational, finite, linear combinations of elements from an arbitrary subset $X$ of $\mathbb{R}\setminus\mathbb{Q}$, and all the multiplicative and additive inverses of any number of combinations.
It is not so simple because if you choose, say, numbers of the form $r+r^\prime\times\pi$, you need also $(r+r^\prime\times\pi)^{-1}$ and so on.
One more possibly interesting point is the fact that there are Hamel basis for $\mathbb{R}$. So, if you want to characterize all subsets of $\mathbb{R}$ which fulfill all ordered field axiom but not the completeness, you would need to require that the set $X$ cannot include any Hamel basis, etc.
A: The completeness axiom in the axiomatization of the reals is not so much like Euclid's Fifth. The completeness axiom is, well, a property that assures completeness. Euclid's Fifth has nothing to do with completeness. Further, there are many non-isomorphic models for geometry with Euclid's Fifth (e.g., the Euclidean spaces). However, all models of the axioms of the reals (as a complete ordered field) are isomorphic. Negating Euclid's Fifth admits new types of geometry that make perfect sense geometrically. Negating the completeness axiom of the reals results in many models (e.g., the rationals) that are quite terrible for the purposes of analysis. In geometry there are many consequences of Euclid's Fifth as well equivalent formulations in terms of equally reasonable geometric notions (e.g., triangles having angles summing up to $\pi$ radians). In analysis the completeness axiom has a few equivalents, but they are rather set-theoretic.
So, to see some of the bad things that happen if you neglect the real's completeness can be seen from trying to do analysis in $\mathbb Q$. Say bye-bye to $\sqrt 2$ and many other important numbers like $\mathrm e,\pi $, etc. Of course, a trace of these numbers is still present, namely instead of $\sqrt 2$ you have sequences of rationals that approach $\sqrt 2$. Of course, any such sequence of rationals that approximate $\sqrt 2$ is as good as any other, so you'd really like to treat all these sequences as equivalent. And of course you'd like to repeat that process for any 'number' that is missing in your system but can be approximated from within the system. Oh, but some sequences don't approximate anything at all. How can we detect those that do? Of course, they all satisfy Cauchy's condition (spelled out only using rationals). So, let's take all the Cauchy sequences, mod out by the suitable equivalence relation, and, voilà, we've just constructed the reals.
In other words, from the perspective of analysis, you always want to work with a complete space simply since it's convenient. The only thing that will change if the space is not complete is that you may run against some entities that you can perfectly well approximate within your system, but that do not have a 'name' within the system. There is a hole in the space. The space is not complete. Completing it merely introduces names for all the holes, so that whatever you can approximate within your system is necessarily in the system. That's all the completion does. It's not really a property of the same nature as Euclid's Fifth is. Euclid's Fifth is saying something about how the geometry behaves. Completeness simply states that the model is rich enough to contain everything it approximates. If that was not the case, you can always apply a generic completion process. There is no Fifthication process in geometry.
