# Category Archives: Set Theory

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 to suggest that you can pick a least number from every set including the reals, so why can’t you just keep picking least elements to put them into 1 to 1 correspondence with the reals. I understand why Cantor says you can’t. I just don’t see what is wrong with the other arguments (other than it must be wrong somehow). Apologies for not being able to state the argument in formal maths, I’m around 20 years out of practice for formal maths.

As we’ve seen in too many discussions of Cantor’s diagonalization, it’s a proof that shows that it is impossible to create a one-to-one correspondence between the natural numbers and the real numbers.

The Well-ordering says something that seems innoccuous at first, but which, looked at in depth, really does appear to contradict Cantor’s diagonalization.

A set $S$ is well-ordered if there exists a total ordering $<=$ on the set, with the additional property that for any subset $T \subseteq S$, $T$ has a smallest element.

The well-ordering theorem says that every non-empty set can be well-ordered. Since the set of real numbers is a set, that means that there exists a well-ordering relation over the real numbers.

The problem with that is that it appears that that tells you a way of producing an enumeration of the reals! It says that the set of all real numbers has a least element: Bingo, there’s the first element of the enumeration! Now you take the set of real numbers excluding that one, and it has a least element under the well-ordering relation: there’s the second element. And so on. Under the well-ordering theorem, then, every set has a least element; and every element has a unique successor! Isn’t that defining an enumeration of the reals?

The solution to this isn’t particularly satisfying on an intuitive level.

The well-ordering theorem is, mathematically, equivalent to the axiom of choice. And like the axiom of choice, it produces some very ugly results. It can be used to create “existence” proofs of things that, in a practical sense, don’t exist in a usable form. It proves that something exists, but it doesn’t prove that you can ever produce it or even identify it if it’s handed to you.

So there is an enumeration of the real numbers under the well ordering theorem. Only the less-than relation used to define the well-ordering is not the standard real-number less than operation. (It obviously can’t be, because under well-ordering, every set has a least element, and standard real-number less-than doesn’t have a least element.) In fact, for any ordering relation $\le_x$ that you can define, describe, or compute, $\le_x$ is not the well-ordering relation for the reals.

Under the well-ordering theorem, the real numbers have a well-ordering relation, only you can’t ever know what it is. You can’t define any element of it; even if someone handed it to you, you couldn’t tell that you had it.

It’s very much like the Banach-Tarski paradox: we can say that there’s a way of doing it, only we can’t actually do it in practice. In the B-T paradox, we can say that there is a way of cutting a sphere into these strange pieces – but we can’t describe anything about the cut, other than saying that it exists. The well-ordering of the reals is the same kind of construct.

How does this get around Cantor? It weasels its way out of Cantor by the fact that while the well-ordering exists, it doesn’t exist in a form that can be used to produce an enumeration. You can’t get any kind of handle on the well-ordering relation. You can’t produce an enumeration from something that you can’t create or identify – just like you can’t ever produce any of the pieces of the Banach-Tarski cut of a sphere. It exists, but you can’t use it to actually produce an enumeration. So the set of real numbers remains non-enumerable even though it’s well-ordered.

If that feels like a cheat, well… That’s why a lot of people don’t like the axiom of choice. It produces cheatish existence proofs. Connecting back to something I’ve been trying to write about, that’s a big part of the reason why intuitionistic type theory exists: it’s a way of constructing math without stuff like this. In an intuitionistic type theory (like the Martin-Lof theory that I’ve been writing about), it doesn’t exist if you can’t construct it.

# From Sets to Groups: Deep Meaning in Simple Constructs

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 of that is something
called a group. Once we’ve built up sets enough to be able to understand a set of values and an operator, we’re able to build an amazing deep and interesting construction, called a group.

Back when I started this blog, one of the first topics that I
wrote about was group theory. I was just looking back over that
series of posts, and frankly, they sort of stink. I’ve leaned a lot about
how to write for a blog in the time since then, and so I’d like to go back
and rewrite it. I’ve never reposted any of the group theory material, so
it should also be new to most readers.

# From Sets to Numbers: Climbing Up to the Rationals

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
up the full tower of numbers, showing how if we’ve got the natural numbers defined in sets, we can define
the rational numbers using sets and our constructed integers.

# From Sets to Arithmetic

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 take a bit of time to go through some basic constructions using set theory.

We’ve seen, roughly, how to create natural numbers using nothing but sets – that’s basically what
the ordinal and cardinal number stuff is about. Even doing that much is tricky – witness my gaffe about
ordinals and cardinals and countability. (What I was thinking of is the difference between the ε series in the ordinals, and the ω series in the cardinals, not the ordinals and cardinals themselves.) But if we restrict ourselves to sets of finite numbers (note: sets of finite numbers, not finite sets of numbers!), we’re pretty safe.

Of course, we haven’t defined arithmetic – we’ve just defined numbers. You might think it would be
pretty important to define arithmetic on the numbers. If you thought that, you’d be absolutely
Correct. So, that’s what I’m going to do next. First, I’m going to define addition and subtraction – multiplication can be defined in terms of addition. Division can be defined in terms of multiplication
and subtraction – but I’m going to hold off on defining division until we get to rational numbers.

# Tiptoeing into Type Theory

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 that we’ve looked at (ZFC and NBG), there were attempts by changing the basis of set theory – discarding sets in favor of something similar, but with restrictions that avoid the problems of naive set theory. Today, I’m going to talk about an example of the latter approach, called type theory. Type theory is a very different approach from what we’ve seen before, and one which is particularly useful and interesting to people in my neck of the woods.

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 to a few more posts about the beautiful world of set theory.

When you talk about set theory, you’re talking about an extremely abstract notion, one which is capable of representing all sorts of structures: topological spaces, categories, geometries, graphs, functions, relations, and more. And yet, almost every description of set theory plunges straight into the cardinal and ordinal numbers. Why? That’s a question that mystified me for quite a long time. Why do we take this beautiful structure, which can do so many things, and immediately jump in to these odd things about infinities?

# Graph Searches and Disjoint Sets: the Union-Find Problem

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 new vertices and edges being added to the graph, but nothing is ever removed. Some questions you might want to ask about this graph at a particular point in time are:

• How many components are there in the graph?
• Which component is vertex X in?
• Are vertices X and Y in the same component?
• How many components are there?

All of these questions are variants of a classic computer science problem, called
union-find, which also comes up in an astonishing number of different contexts. The reason for the name is that in the representation of the solution, there
are two basic operations: union, and find. Basically, the division of the graph into
components is also a partition of the vertices of the graph into disjoint sets: union find
is a problem which focuses on a particular kind of disjoint set problem, where you can modify
the sets over time.

# Alternative Axioms: NBG Set Theory

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 day that the ordinals are a proper class is an NBG proof.) NBG is an alternate formulation of set theory which has the same proof power as ZFC, but does it with a finite set of axioms. (If you recall, several of the axioms of ZFC are actually axiom schemas, which need to be distinctly instantiated for all possible predicates.) NBG uses one axiom scheme, but it’s possible to show that that schema only expands into a finite number of distinct axioms.

# Ordinal Exponents and Really Big Numbers

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 we
can’t express in terms of any finite sequence of multiplications.

# More on Ordinals: Ordinal Arithmetic (part 1)

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.