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 elements of a set can be put in order, even if it’s infinite: how do we talk about *where* an element occurs in a well-ordering of an infinite set?
For doing this, there’s a construction similar to the cardinal numbers called the *ordinal numbers*. Just like the cardinals provide a way of talking about the *size* of infinitely large things, ordinals provide a way of talking about *position* within infinitely large things.

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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 transfinite numbers, which form a sequence of ever-larger infinities.
One major problem with adding the transfinite numbers is that natural number arithmetic doesn’t work anymore with the cardinals. It still works for the natural number subset of the cardinals, but not for the transfinites.
But we *do* want to be able to talk about at least certain kinds of arithmetic on the full set of cardinals. So we need to figure out what arithmetic means for this strange sort of number.

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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 its cardinality. But there are interesting questions that we can ask about the relative size of different sets, even when those sets have an infinite number of elements. And that’s where things get really fun.

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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 called the well-ordering theorem, which is fully equivalent to the axiom of choice. What it says is that every set has a well-ordering. Which doesn’t say much until we define what well-ordering means. The reason that it’s so important is that the well-ordering theorem means that a form of inductive proof, called transfinite induction can be used on all sets.

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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 the objects of S(O), and them prove the axioms A about S(O) using the axioms of ZFC.

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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 can create two spheres of size S out of one sphere of size S cutting the single sphere into pieces, and then gluing those pieces back together. Volume from nowhere, and your spheres for free!

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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 of it, it’s difficult to understand how it managed to cause so much trouble. For example, wikipedia has a rather nice informal statement of it:

given a collection of bins each containing at least one object, exactly one object from each bin can be picked and gathered in another bin

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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 get the integers; with the integers, we can get the rationals. Once we have the rationals, things get a bit harder – but we can get the reals via Dedekind cuts; and by transfinite induction, we can get the transfinite numbers. But before we can get to any of that, we need a sound representation of the naturals in terms of sets.

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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 equality means. It says that two sets are equal if, and only if, they have the same members: that is, a set is completely determined by its contents.

How much more trivial can a statement about sets get? It really doesn’t seem to say much. But what happens when we start thinking through what that means?

The way we normally think of sets, they’re collections of objects. So, imagine a set like {red, green, blue}, where the three values are atoms: that is, they’re single objects, not collections. What does the axiom of extensionality say about that? It says that red, green, and blue are not atoms?

Why? Well – let’s look at the axiom of extensionality again: (∀A,B: A=B ⇔
(∀C: C∈A ⇔ C∈B)). So – does red = blue? Well, if they’re atoms, then
yes, red=blue, because nothing is in red, and nothing is in blue. Since neither has any members, they’re equal.

In fact, if we follow that reasoning through, there’s only one possible atom: the only set with no members is the empty set. So anything else we want, we’re going to have to represent using some kind of collection.

As a result of that, along with the axiom of specification, we can show that the axiom
of the empty set is actually redundant. After all – the axiom of specification basically
says that if you can describe a collection of values by a predicate, that collection is a class. So take the predicate P(x)=false; that’s a set with no values. Also known as the empty set. So the empty set exists, and it’s the only set with no members – aka the only atom.

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 cartesian product, huge ranges of interesting and
important areas of mathematics would be inaccessible to us. (Note that I’m saying that
pairing is necessary, not that it’s sufficient. You also need replacement
to get the projection functions that are part of the usual definition of the cartesian
product.)

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