Set Theory:
The point of set theory isn't just to sit around and twiddle our thumbs about the various definitions we can heap together. It's to create a basis on which we can build and study interesting things. A great example...
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Posted on December 3, 2007 8:21 AM • 69 Comments •
When last we left off, I'd used set theory to show how to construct the natural numbers; and then from the natural numbers, I showed how to construct the integers. Now, we can take the next step in building...
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Posted on November 29, 2007 3:54 PM • 13 Comments •
Even though this post seems to be shifting back to axiomatic set theory, don't go thinking that we're done with type theory yet. Type theory will make its triumphant return before too long. But before that, I want to...
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Posted on November 19, 2007 2:53 PM • 34 Comments •
When Cantor's set theory - what we now call naive set theory - was shown to have problems in the form of Russell's paradox, there were many different attempts to salvage the theory. In addition to the axiomatic approaches...
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Posted on November 15, 2007 10:55 AM • 18 Comments •
I was visiting my mom, and discovered that I didn't leave my set theory book on the train; I left it at her house. So I've been happily reunited with my old text, and I'm going to get back...
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Posted on November 11, 2007 8:45 PM • 20 Comments •
Suppose you've got a huge graph - millions of nodes. And you know that it's not connected - so the graph actually consists of some number of pieces (called the connected components of the graph). And there are constantly...
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Posted on September 6, 2007 11:40 AM • 12 Comments •
So far, we've been talking mainly about the ZFC axiomatization of set theory, but in fact, when I've talked about classes, I've really been talking about the von Newmann-Bernays-Gödel definition of classes. (For example, the proof I showed the other...
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Posted on June 21, 2007 6:44 PM • 10 Comments •
With ordinals, we use exponents to create really big numbers. The idea is that we can define ever-larger families of transfinite ordinals using exponentiation. Exponentiation is defined in terms of repeated multiplication, but it allows us to represent numbers that...
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Posted on June 17, 2007 3:22 PM • 27 Comments •
I'll continue my explanation of the ordinal numbers, starting with a nifty trick. Yesterday, I said that the collection of all ordinals is not a set, but rather a proper class. There's another really neat way to show that....
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Posted on June 13, 2007 10:00 PM • 8 Comments •
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...
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Posted on June 12, 2007 9:31 PM • 6 Comments •
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...
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Posted on June 11, 2007 9:46 PM • 12 Comments •
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,...
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Posted on June 10, 2007 9:22 PM • 16 Comments •
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...
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Posted on June 7, 2007 4:47 PM • 38 Comments •
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...
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Posted on June 6, 2007 9:15 AM • 39 Comments •
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...
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Posted on June 4, 2007 9:33 PM • 33 Comments •
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...
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Posted on May 27, 2007 5:43 PM • 63 Comments •
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...
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Posted on May 24, 2007 9:43 PM • 16 Comments •
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...
Posted on May 23, 2007 9:43 PM • 12 Comments •
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...
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Posted on May 23, 2007 2:50 PM • 9 Comments •
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...
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Posted on May 20, 2007 10:08 PM • 34 Comments •
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...
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Posted on May 19, 2007 8:22 PM • 37 Comments •
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...
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Posted on May 16, 2007 7:02 PM • 26 Comments •
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...
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Posted on May 14, 2007 9:56 PM • 42 Comments •
