Mark Chu-Carroll is a Computer Scientist working as a researcher in a corporate lab. My professional interests run towards how to build programming languages and tools that allow groups of people to work together to build large software systems.
goodmath
Posts by this author
June 12, 2007
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…
June 11, 2007
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…
June 10, 2007
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…
June 7, 2007
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…
June 6, 2007
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…
June 5, 2007
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?
June 4, 2007
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…
June 3, 2007
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…
June 2, 2007
The 9th Carnival of Mathematics is live at JD2718, in a cleverly alphabetical form.
June 1, 2007
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.…
May 31, 2007
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…
May 28, 2007
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.…
May 27, 2007
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…
May 24, 2007
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…
May 24, 2007
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:…
May 23, 2007
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…
May 23, 2007
The axiom of pairing is an interesting beast. It looks simple, and in fact, itis 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 cartesian…
May 22, 2007
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…
May 20, 2007
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…
May 19, 2007
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…
May 17, 2007
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…
May 16, 2007
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…
May 16, 2007
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…
May 14, 2007
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,…
May 14, 2007
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…
May 13, 2007
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…
May 11, 2007
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…
May 11, 2007
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…
May 11, 2007
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…
May 10, 2007
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…