I am reading Principles of Mathematical Analysis by Rudin. Theorem 1.19 in the book says that
"there exists an ordered field $\Bbb{R}$ which has the least-upper-bound property".
Then author have called members of $\Bbb{R}$ as real numbers. As the proof was bit long, author gave it at the end of chapter. I too skip the proof for some time and went on reading further theorems assuming $\Bbb{R}$ to be our real line but now when I look the proof, real field $\Bbb{R}$ constructed by author is completely different. The members of $\Bbb{R}$ are some subsets of $\Bbb{Q}$ (he called them cuts). I got what author did and his construction but now when I look again at the theorems 1.20 and 1.21 with author's newly constructed field $\Bbb{R}$ ( which I was assuming our real line), they are not making much sense to me.
I am somehow confused all of a sudden what exactly going on. Can someone please help to explain author's constructed real field $\Bbb{R}$ (whose members are cuts). Is this same as our normal real field (whose members are numbers)?
Best Answer
Yes, it is the same. Cuts are numbers.
There are many different ways to construct the real numbers from the rational numbers (two popular constructions in particular, but I'm sure there are more exotic ones out there). Using so-called Dedekind cuts is one. Regardless of the construction chosen, the elements of the construction will not "look like" numbers (whatever that means). But they are numbers nonetheless.
A cut is a non-empty, upwards-bounded subset $X$ of the rational numbers with no maximal element, where each element of $X$ is a lower bound for the complement of $X$. (Some define a cut to be the pair consisting of $X$ and the complement of $X$.) This is defined to be the real number that intuitively corresponds to the least upper bound of $X$. So the cut construction of the real numbers intuitively takes $\Bbb Q$, and "injects" least upper bounds to all sets that should have least upper bounds but don't.
Note that different constructions will give real numbers that look different. However, this is mostly irrelevant, as we rarely care about what real numbers are, we care about how they behave (the same goes for a lot of other mathematical structures as well). Basically the only reason to have a concrete construction (like the cuts) is to demonstrate that a set of numbers with the properties we would like the real numbers to have, actually exists.
Once we have shown that a construction, like the cuts, gives us a set of numbers that have the properties of what we call "real numbers", we forget all about what real numbers are, and change our focus to what properties they have, like an order, least upper bounds, a notion of addition and multiplication, and so on.