Can someone help me understand the concept of Dedekind cut? I'm having trouble understanding it because it appears that in order to understand what Dedekind cuts are I already must have a very developed idea of real numbers. For example, the picture of a cut is the set of rationals smaller than a real number. Also, when I examine the definitions of addition and multiplication, they seem kind of like randomly constructed definitions, I don't think "yes, this definition really defines the addition". It seems I have to prove that the definition corresponds to some concept of addition and multiplication of real numbers that is already very understood. Thanks.
There are all sorts of things going on here, so it is difficult to untangle them.
We do have an intuitive idea of the real numbers as being "the complete number line with no gaps". We know that this isn't the rational numbers because they don't include $\sqrt 2$. But we have also an idea that the rationals are dense, so we ought to be able to get as close as we want to any number on the line by choosing a suitable rational number. Actually we notice that any number $x$ on the line divides the rational numbers into two disjoint sets - the rational numbers $\le x$ and the rational numbers $\gt x$. And that is all the rationals. Different points on the line give different sets of rationals.
With that intuition in mind, we turn the idea around. If we have two disjoint sets of rationals $L$ and $R$ which between them contain all the rationals and where every member of $L$ is less than every member of $R$ (and depending on treatment we might say that $R$ has no least element), we imagine there might be a gap between $L$ and $R$, so we need a number to fill the gap which we might call $L:R$. In this way we have defined some new numbers on the line which fill in all the gaps.
But are they numbers? We know that the rational numbers obey the basic laws of arithmetic (the axioms for a Field). And each of the gaps we've identified is very close to a lot of rational numbers, so we hope they will work for the new numbers - but we need to check.
Then the numbers are on a line, so we need to check they are properly ordered, so that our intuition about the line is satisfied.
Then we were worried about gaps in the line - have we managed to fill them all? We need to check that the new numbers are complete.
So we show that our construction gives us a complete ordered field - the Real numbers. We could have given an abstract definition of such a thing, and hoped that it would exist. The Dedekind construction shows that it does - in essence it is an existence proof - it tells us that we can choose our system of numbers to have the good properties we wanted it to have.
In another part of the forest we can show that there is, up to isomorphism, only one complete ordered field - a proof of uniqueness - so when we talk about Real numbers, all our definitions and constructions are equivalent. That too is very useful - we know what we are talking about.
I'm going to give a slightly different perspective to try to answer the underlying question. Dedekind cuts aren't "the real numbers", as if there were only one collection of numbers that can fulfill that role. What Dedekind cuts do is to give one of many different ways to construct a collection that has all the properties of real numbers that had been previously 'known' or rather assumed. For example we could use axioms to stipulate real arithmetic and the 'fact' that any set of real numbers with an upper bound has a least one. However, how do we know that there really is such a collection of objects satisfying the axioms? That last 'fact' is in fact not one that is obviously true. But if we accept the existence of the field of rational numbers, then we can indeed construct its completion which we can then prove to satisfy that fact.
Besides Dedekind cuts, another way to construct 'real numbers' is to use Cauchy sequences of rationals, which may or may not seem more natural in the sense that we generally do use such approximation sequences to approximate real numbers in the real world. Furthermore, the operations we need to define on these Cauchy sequences to get arithmetic and supremum are natural as well.
If constructing 'real numbers' is the only goal, then any way would do, because we can subsequently prove that any such collection is essentially unique, meaning that any two such collections are isomorphic, and furthermore that the isomorphism itself is unique. This means that we don't gain or lose anything at all by using one construction over another, and hence we can call any such collection as "the real numbers".
However, Dedekind cuts do have something to offer that Cauchy sequences don't. They can be used to complete any total order, although there is no arithmetic. On the other hand, Cauchy sequences are more constructive in a certain sense. There is an interesting question at Math Overflow on a set theoretic model where they are different.
Consider the subset of Dedekind cuts in which the left-hand cut has a greatest element (or right-hand cut has a least element, depending on whose definition you used). That is, consider the cuts that correspond to rational numbers. You should be able to make definitions of addition and multiplication of these cuts that match the existing definitions of addition and multiplication of rationals. That is, it makes sense to define addition and multiplication of cuts in such a way that the sum of the cuts corresponding to rational numbers $p$ and $q$ is the cut corresponding to $p + q$, and the product of those two cuts is the cut corresponding to $pq$.
If you do that, then the definitions of addition and multiplication of irrational cuts should make sense.