Mathematics
This is a remnant of my old site. It contains various pieces of mathematics I had been working on. I don’t think anything here has been updated in the best part of two years.
Writings
These are my major writings on mathematics, both completed and works in progress. There’s nothing new and revolutionary here: They’re all things which I’ve written to introduce or clarify a subject.
The Set Theoretic Structure of Analysis
My Part III essay. It looks at some of the ways in which analysis depends on the underlying set theoretic axioms, principally the implications of the axiom of choice, the continuum hypothesis and various weakened forms and negations of these. Warning: Some of the proofs in this are wrong, and I never got around to correcting it.
Filters in Analysis and Topology
An introductory article on filters as a method of talking about convergence in metric and topological spaces. Introduces basic ideas and shows how they can be applied in familiar settings, among other things providing a nice proof of Tychonoff’s theorem.
Some Cute Analysis Problems
A collection of analysis problems which I have gathered. Some are my own, some are standard or the result of problems and or questions I’ve recieved from friends or seen on the web. They all have elementary solutions, but may well require some thought!
Counting to Infinity: Ordinals and Transfinite Processes
Some notes I wrote up to clarify and expand upon an online seminar I gave with the same title. It’s an introduction to the basic notions of ordinals and transfinite induction / recursion for non set theorists. It assumes the working knowledge of set theory that you would expect a typical mathematician to have (say the equivalent of the Cambridge mathematical tripos’s IA numbers and sets course) and a little bit of algebra, topology and analysis. The latter three aren’t neccesary to follow all of it, but without some of them you’ll miss a lot of the motivating examples. Here is the log of the original seminar, but you’re probably better off reading the notes.
Analysis in the World According to David
At one point I was writing an analysis textbook, trying to teach it in a somewhat different way than it is usually taught. It didn’t really go anywhere, but this is an old draft of it.
Some Theorems
A collection of random theorems which I’ve written up proofs of for one reason or another.
Cardinal Arithmetic via the Upward Lowenheim Skolem Theorem
A silly proof of some basic results in cardinal arithmetic.
A Continuous Surjection from the Cantor Set
A proof of the classic result that every non-empty compact metric space is the continuous image of the cantor set.
Some Lemmas
A collection of random lemmas, mostly on convergence theorems about real numbers or real valued functions. It’s all fairly elementary stuff, which I’ve included either because it amused me, because the proof is slightly tricky, or because it’s somewhat obscure and I wanted a record of it. Or another reason. It’s currently rather empty, but will increase.
