The Axiom of Choice The axiom of choice is a fascinating bugger. It’s probably the most controversial statement in mathematics in the last century – which is pretty serious, considering the kinds of things that have gone on in math during the last century. The axiom itself is quite simple, and reading an informal description […]
As you may have noticed, E. E. Escultura has shown up in the comments to this blog. In one comment, he made an interesting (but unsupported) claim, and I thought it was worth promoting up to a proper discussion of its own, rather than letting it rage in the comments of an unrelated post. What […]
One of the reasons that the axiom of choice is so important, and so necessary, is that there are a lot of important facts from other fields of mathematics that we’d like to define in terms of set theory, but which either require the AC, or are equivalent to the AC. The most well-known of […]
Today, I’m going to try to show you an example of why the axiom makes so many people so uncomfortable. When you get down to the blood and guts of what it means, it implies some *very* strange things. What I’m going to do today is tell you about one of those: the Banach-Tarski paradox, […]
Axiomatic set theory builds up set theory from a set of fundamental initial rules. The most common axiomatization, which we’ll be used, is the ZFC system: Zermelo-Fraenkel with choice set theory. The ZFC axiomatization consists of 8 basic rules which are pretty much universally accepted, and two rules that are somewhat controversial – most particularly […]
Today’s basics topic was suggested to me by reading a crackpot rant sent to me by a reader. I’ll deal with said crackpot in a different post when I have time. But in the meantime, let’s take a look at axioms.
The other day, I received an email that actually excited me! It’s a question related to Cantor’s diagonalization, but there’s absolutely nothing cranky about it! It’s something interesting and subtle. So without further ado: Cantor’s diagonalization says that you can’t put the reals into 1 to 1 correspondence with the integers. The well-ordering theorem seems […]
As usual, I’m going in several different directions. I’m going to continue doing data structure posts, but at the same time I also want to get back to the type theory stuff that I was writing about before I put the blog on hiatus. So let’s get back to a bit of Martin-Loff type theory! […]
I’ve written about intuitionistic logic before. In fact, there’s a whole section about it in my book. But now that I’m reading a lot about type theory, I’m starting to look at it diferently. When you study classical axiomatic set theory, you’re necessarily also studying classical first order predicate logic. You have to be doing […]
This post started off as an introduction to a post about the simply typed lambda calculus. It got a bit out of control – but reading it over, I think that it’s valuable background. So I’ve made it into its own post. ST type theory is an interesting idea, but it’s mostly interesting as a […]