## The Cardinality of the Continuum

February 24, 2010

A nice grand title to pique your interest!  After some thought and a couple of conversations, I have decided to keep this blog very much aimed at the layman; the thinking being that I don’t particularly want to write hard maths in my spare time, mathematicians don’t particularly want to read hard maths in their spare time, and non-mathematicians definitely don’t want to read hard maths ever.

My PhD supervisor recently appeared on a BBC programme about infinity, which, while good viewing, was rather over-ambitious, and so had to skip over some interesting stuff.  I thought I’d fill in some of the gaps in this post.  So what is this continuum?  Technically a continuum can be anything that is continuous – that is it goes through smooth, infinitesimally gradual transitions, and has no discontinuities or “jumps”.  But in reality the word is rarely used outside of Star Trek and mathematics.  The continuum I will write about is not the space-time continuum, but the real numbers,$\mathbb{R}$.  In my post on$p-$adic numbers, I mentioned that completeness is an important mathematical attribute for a space of numbers to possess.  The notion of continuity is even more fundamental, and people often refer to mathematics as having 2 distinct branches: continuous mathematics, and discrete – or discontinuous – mathematics.