# Gracious Living

The Construction of the Reals: Metric Completion and Dedekind Cuts
March 6, 2011, 21:31
Filed under: Algebra, Math, Set Theory, Topology | Tags: , , , ,

It looks like I’m getting views now, which is surprising.  I’ve been pretty busy with schoolwork, but I really want to get this blog up to speed, particularly because I’d like to start discussing things as I’m learning about them.  I’d also like to make more non-mathematical posts, but maybe these are best left to a separate blog?  Thoughts?

Our first example of a field was the field of rationals, $\mathbb{Q}$.  Recall that this was the field of fractions of the integers, which were in turn the free abelian group on one generator with their natural multiplication.  But now it appears that we’re stuck.  While we intuitively know what $\mathbb{R}$ should be — it’s a line, for crying out loud — there seems to be no algebraic way of “deriving” it from $\mathbb{Q}$.  A first guess might be to add in solutions of polynomials, like $\sqrt{2}$ as the solution of $f(x)=x^2-2$, but not only does this include some complex numbers, it also misses some real numbers like $e$ and $\pi$.  (We call such numbers — those that aren’t solutions of polynomials with rational coefficients — transcendental.  It’s actually quite difficult to prove that transcendental numbers even exist.)

Instead, we turn to topology.  Below, I give two ways of canonically defining $\mathbb{R}$, one using the metric properties of $\mathbb{Q}$, one using its order properties.  I found this really interesting when I first saw it, but I can’t see it interesting everyone, so be warned if you’re not a fan of set theory or canonical constructions.  One of the topological techniques we’ll see will be useful later, but at that point it’ll be treated in its own right.