goodmath

I've talked about the idea of the size of a set; and I've talked about the well-ordering theorem, that there's a well-ordering (or total ordering) definable for any set, including infinite ones. That leaves a fairly obvious gap: we know how big a set, even an infinite one is; we know that the…

This is a short post, in which I attempt to cover up for the fact that I forgot to include some important stuff in my last post. As I said in the last post, the cardinal numbers are an extension of the natural numbers, which are used for measuring the size of sets. The extended part is the…

One of the strangest, and yet one of the most important ideas that grew out of set theory is the idea of cardinality, and the cardinal numbers. Cardinality is a measure of the size of a set. For finite sets, that's a remarkably easy concept: count up the number of elements in the set, and that's…

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 these is…

One of the things that we always say is that we can recreate all of mathematics using set theory as a basis. What does that mean? Basically, it means that given some other branch of math, which works with some class of objects O using some set of axioms A, we can define a set-based construction of…

So today's my thirteenth wedding anniversary. And what did my lovely wife buy me as a present? Yes, a Klein bottle coffee mug. Does she know me well, or what?

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, in which you…

Apparently William Dembski, over at Uncommon Descent is *not* happy with my review of Behe's new book. He pulls out a rather pathetic bit of faux outrage: "Are there any anti-ID writings that the Panda's Thumb won't endorse?" The outrage really comes off badly. But what's Debski and his trained…

The 9th Carnival of Mathematics is live at JD2718, in a cleverly alphabetical form.

All week, I've been buried by a wave of requests to write about LOLCODE today. Normally, I do try to honor requests from readers, but from the time I started my friday pathological languages, I've always tried to stick to languages that actually had *something* interesting about their semantics.…

I've gotten my hands on a review copy of Michael Behe's new book, "The Edge of Evolution". The shortest version of a review is: Bad science, bad math, and bad theology, all wrapped up in a pretty little package. As people who've followed his writings, lectures, and court appearances know, Behe is…

Today is Memorial Day, and I feel compelled to say something about it. We're in the middle of a horrible and pointless war. A war that we started, based on a bunch of lies. Since we did this, we have caused the deaths of hundreds of thousands of innocent Iraqis, and thousands of American soldiers.…

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…

The axiom of infinity is a bundle of tricks. As I said originally, it does two things. First, it gives us our first infinite set; and second, it sets the stage for representing arithmetic in terms of sets. With the axiom of infinity, we get the natural numbers; with the natural numbers, we can…

We've been having some problem with popups showing up on scienceblogs. It's not deliberate: Seed does not accept popup ads. But it appears that one snuck in somehow, but no one is sure where it's coming from. If you see a popup on GM/BM, please do me a favor, and post a comment here telling me…

Some of the basic axioms of ZFC set theory can seem a bit uninteresting on their own. But when you take them together, and reason your way around them, you can find some interesting things. Let's start by looking at the axiom of extensionality. Pretty simple, right? All it does is define what set…

The axiom of pairing is an interesting beast. It looks simple, and in fact, it is simple. But it opens up a range of interesting things that we'd like to be able to do. For example, without the axiom of pairing, we wouldn't be able to formulate the cartesian products of sets - and without…

My father died on sunday. To some degree, I'm still in shock. Even though we knew it was coming, when something like this happens, no amount of preparation really helps. He'd been sick with an antibiotic resistant infection since November, and on thursday, refused to let them give him a feeding…

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…

Naive set theory is fun, and as we saw with Cantor's diagonalization, it can produce some incredibly beautiful results. But as we've seen before, in the simple world of naive set theory, it's easy to run into trouble, in the form of Russell's paradox and a variety of related problems. For the…

I've been seeing articles popping up all over the place about a recent PLOS article called Order in Spontaneous Behavior. The majority of the articles seem to have been following the lead of the Discovery Institute, which claims that the article demonstrates the existence of free will, which they…

So, what's set theory really about? We'll start off, for intuition's sake, by talking a little bit about what's now called naive set theory, before moving into the formality of axiomatic set theory. Most of this post might be a bit boring for a lot of you, but it's worth being a bit on the…

Have you ever wondered about the real reason why math education in our schools is so awful? Why despite the best efforts of large numbers of parents, the schools seem to be incapable of figuring out why they're so dreadfully bad at recognizing the difference between a halfway decent math…

While I've been writing about the Surreal numbers lately, it reminded me of some of the fun of Set theory. As a result, I've been going back to look at some old books. Since I've been enjoying it, I thought you folks would as well. Set theory, along with its cousin, first order predicate logic,…

I debated about whether or not I should write this post. But as you can see, in the end, I overcame my better judgement, and so he we are. Over the weekend, PZ wrote a Pharyngula post about the reaction people have had to Mitt Romney's statement about evolution. He was pissed. And I agree with…

Sorry for the slow pace of things around here lately; life interferes sometimes. I've mentioned my fathers illness before; things took a drastic change for the worse a week ago, which made things more than a little bit crazy. Of course, life wouldn't be life in things happened one at a time; so…

This isn't math, but I felt like commenting anyway. That shining example of an Intelligent Design advocate, Dr. Michael Egnor, is back once again. And this time, his point, such as it is, is to basically fling insults at PZ Myers. What did PZ do to bring on his ire? Well, PZ was annoyed with Time…

Baka Beyond, "Baka Play Baka": This is what happens when you take a bunch of great trad Irish musicians, and lock them into a room with a bunch of great African musicians from the Baka tribe in Cameroon. I don't know quite how to describe this. It really doesn't sound like anything else. You can…

Today's a mighty cool example of bizzare language design, called GammaPlex In terms of language design, it's nothing particularly special: it's yet another stack language with a befunge-like graphical syntax. What's unusual about GammaPlex is that it's strongly focused on graphics. It's got built…

In the comments on my DMCA post, a reader asked me to comment on this piece of silliness. I try not to disappoint my readers, so here's my take. It's a pile of silliness with the distinct aroma of astrotur - silliness mixed with a bit of deliberate stupidity in order to obscure things. The basic…