Dedekind Cuts, Big Picture

You might have heard about statements like “Dedekind cuts is a rigorous way to construct the reals,” or “Dedekind cuts are the reals,” etc. But in my opinion, to truly understand Dedekind cuts, you must first forget about these statements, learn what Dedekind cuts are—directly and literally from its formal definitions, and then come back to look at how it connects to the reals. Usually, mathematical concepts make more sense when you learn about the context along with the technical details; yet for Dedekind cuts, having contextual knowledge at the back of your head as you learn the formal definitions and related proofs just makes everything even more confusing.

So let us start from the basics.

We know

I think today's material is better understood by thinking analogously to vector spaces. If you recall, a vector space consists of a Domain, a Vector Addition operator, and a Scalar Multiplication operator. An example---we first define:

• Domain: {(x,y) |x,y ∈ Q} i.e. the set of rational pairs
• Vector Addition: (x,y) + (a,b) = (x+a,y+b)
• Scalar Multiplication: c(x,y) = (cx,cy)

We do something similar for fields. Just like we deployed the notion of Q and constructed the vector space Q^2, this time we want to deploy, still, the notion of Q, and construct an interesting ordered field, called R. I will explain the philosophical "why we do" in a moment, but for now just focus on the "what we do"---we first define the Domain, Addition, Multiplication, Order, etc. of R, as follows: (remember, the ultimate goal of this discussion is to construct R, which means at this point we should not beg the question and assume R already existed, and so the following definitions avoid deploying the notion of R)

• Domain: α ⊂ Q | a is a Dedekind Cut i.e. the set of Dedekind Cuts (a Dedekind Cut is a subset of Q that has specific properties)
• Addition: α+β = x+y | x in a, y in b
• Multiplication: ...
• ... etc. (as defined in class)

After all of that, we simply define R:=R, with numbers in R just being shorthand notations for Dedekind Cuts in R. For instance, the (true) statement about R:
circled1 0 + 1 = 1
can be contracted to:
circled2 0 + 1 = 1
Or we can also say, when we write down statements about R, such as circled2, it is actually something like circled1 in disguise.

The reason why we go through all of these cumbersome tasks and define R in such a complicated way is that, philosophically we want to make as few assumptions as possible, especially about what kinds of structures exist. Recall the natural numbers. (HAVE NOT FINISHED REVISING THE REST)

the fact that there is actually a structure satisfying the Peano Axioms of Natural Number is not obvious at all. So the existence of N is really just an assumption---we presuppose the existence of N and then do arithmetics. We HAVE TO accept that presupposition, otherwise we cannot do any math.

Now you might think, we also HAVE TO "assume" that there is a structure satisfying the Axioms of Reals (i.e. (P1)-(P13)), but the genius thing about Dedekind Cuts is that, we don't! As we have shown, by only assuming the existence of N, we can then so to speak "construct (no longer assuming)" the existence of structures like Z, Q, and (using Dedekind Cuts) R. In short, Dedekind Cuts allow us to not "assume" the existence of R, but "construct" it!