Now there’s completeness and then there’s completeness…

15 October 2009

This post achieves a fortuitous segue from the last post into my serious of articles on the beauty of Galois theory.

In the previous post I introduced Dedekind cuts as a means of constructing the real number line, and I said that this perspective is responsible for the completeness of the real numbers $\mathbb{R}$.

Now, that was completeness in the topological sense.  There is another, very different notion of algebraic completeness.

A number system is called algebraically complete if every polynomial equation in one variable with coefficients from that number system can be solved in that number system.

I am currently teaching a course in geometry for teachers (Euclidean, nonEuclidean, projective, the whole ball of wax), and we were recently discussing the need for the Dedekind axiom for plane geometry, which guarantees in effect that the points on a geometric line behave the  same , way as real numbers on a real number line.  What was interesting to me was that, even after all we’d said about all the ways that geometry might behave in unexpected ways if we don’t make certain assumptions, somehow the idea that geometric lines were real number lines was more deeply ingrained.  The idea that there might be a world were there were no line segments with length $\pi$ was harder to imagine than the idea of a world where there are multiple lines through a point parallel to another point.