The Axiom of Choice

April 22, 2010

Yes…I’m still alive and blogging.  Just been very busy, as it is coming up to exam-time.  Thankfully I think I can safely say that my exam-sitting days are behind me, but there are a lot of anxious students happy to pay good money to someone who will help them pass theirs!  Anyway, I recently read an interesting blog post about the axiom of choice, and I thought it would be a good topic to give my usual treatment (that is an attempt to explain it to an interested and patient non-mathematician).

Now, unlike other sciences – in which theories and models are often proposed, used for while, and then modified or just plain discredited by subsequent discoveries – mathematics is constantly building on what has gone before.  If sciences were buildings, then mathematics would be a vast complex structure that is constantly being added to: sometimes the foundations would be strengthened, sometimes an extension or a whole new wing would be added, but nothing would ever be demolished.  Physics, on the other hand, would be a lot of separate buildings, some of which have long ago been abandoned, and most of the rest of which are in a constant state of being torn down and rebuilt in the modern style!  This is part of what drew me to the subject in the first place; the fact that if you prove something mathematically, it is definitely true, and will never be disproved.  We still use theorems proved by people that lived thousands of years ago, and it is reassuring to know that anything we prove now will still be being used by people in thousands of years.

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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 onp-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.

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