Constructing the Reals

You wouldn't think that you'd have to jump through too many hoops to construct the Real numbers. However, as I've mentioned before, there's nothing "real" about them. So, one does need to take some care to make sure one knows exactly what we're talking about.

There are two basic approaches to this. The axiomatic approach is to start defining properties that any set that claims to be the real numbers must satisfy. Then one needs to prove that there's exactly 1 set that satisfies those axioms. Two few axioms, you get competing sets. Too many, and you can't satisfy them with anything. Of course, when I say that exactly 1 set satisfies them, I mean that any set that satisfies them has a homeomorphic mapping to the reals. A homeomorphic mapping is one where we just change the names, but all the relationships are left in tact. For example, if we decided to call the additive identity element "A" instead of "0", we'd still have the same set, the elements would just be labeled differently. (Odd side note: the only two elements that are actually "named" in the axioms are the identity elements 0 and 1. These are also the two digits that are easily confused with letters. It's almost like they wanted to drive home the whole homeomorphic thing.)

The axiomatic approach is what I got in my Analysis class in undergrad.

In Set Theory, we're getting a different construction which I'll call the Algebraic approach. Here, we start with the Rationals which, I guess, are considered intuitively obvious and I won't debate that though, being only one axiom away from the reals, once certainly could. We then show that Cauchy sequences on the field of rationals define a ring and one can find produce an ideal that yields the Reals as a coset.

That's all great except for one small problem. Abstract Algebra is a bit of a blind spot for me. I know the basic concepts, but I've never had a formal course in it and certainly do't have the tools to actually prove stuff beyond an undergrad level.

If I hadn't taken this huge bite of a term project in Evolutionary Programming, I'd use this as the excuse to slam through a text in Abstract Algebra. As I really don't have time for that, I'm just going to have to hang on as best I can. Meanwhile, the deficiency is noted. I think that if I really want to call myself a Mathematician, I need to take (or otherwise master) a grad level course in Abstract Algebra and Analytic Geometry. Pragmatists would call me a fool as neither of those topics has even a remote bearing on my thesis, but I really do believe there are things an "expert" is just supposed to know.