What is set theory for?
In other words, why am I spending so many hours of my life lecturing about it?
Why are you spending so many of your hours listening to the lectures?
Why bother with set theory?
Well, this is an answer that you may very well have originally decided.
The answer to for you, I'll say what my answers to this question are.
Set theory is one of those bits of mathematics that some people, but not though very far
from everybody.
Some people have already met before they went to university.
I was one.
So at our primary school, they taught us some basic set theory in sections, unions, that
kind of thing.
It was trendy for a while.
I thought at the time, you know.
What on earth is this all about?
At that level, set theory was just a way of thinking about logic.
There's a lot more to it.
But when I was a bit older, my father told me about infinity and about how to do arithmetic
with infinities.
And I thought that was amazing.
And that is going to happen in this course.
Anyway, what is set theory for?
And if you haven't met it before, then perhaps you've heard about its philosophical ramifications
about the paradoxes of set theory,
about the ways you can use set theory to establish mathematics.
I'll talk about some of those things as well.
Anyway, what is set theory for?
Historically, the answer is Cantor originally developed it for proving a theorem about Fourier series.
So you try and express a function as a sum of cosines and sines.
Sometimes it works, sometimes it doesn't.
The theorem that Cantor proved - actually it wasn't so much the theorem itself.
He had a set of theorems.
He actually had an infinite transfinite sequence of theorems.
And he needed to talk about that infinite transfinite sequence.
He ended up inventing set theory to do it.
Set theory then provides tools.
So it provided the tool that Cantor needed on that occasion.
It provides other tools as well.
So cardinal numbers.
Cantor discovered these. A cardinal number answers the question:
how many?
The word cardinal comes from a Latin word for a hinge.
So cardinal numbers are the pivotal numbers.
The numbers that
Everything else turns on.
So the most important numbers.
Okay.
So cardinal numbers answer the question how many.
How many fingers are there on my right hand?
There are five.
How many rational numbers are there? Aleph-zero.
Okay.
Ordinal numbers: ordinal numbers answer a different kind of question, they answer
a question about places in an arrangement.
And in English and in many other languages, there are special words to refer to ordinal
numbers.
So George Washington was the first president of the United States of America.
Charles the First was the second King of both England and Scotland.
And so far, the only King of England to be executed for treason.
The infinite ordinal numbers...Now the infinite ordinal numbers are weird.
The thing that is most weird about them is that they don't correspond neatly to the infinite cardinal numbers in
the way that the finite ones do.
Okay, the ordinal numbers - set theorists refer to them all the time.
The ordinal numbers are the backbone of modern set theory.
You hardly hear a set theorist opening their mouths without mentioning an ordinal number.
I exaggerate, but not much.
Okay, so set theory provides tools and set theorists are also very good at doing schemes
of coding.
So representing sort of coding up structures in various kinds of of data structures.
Yeah, well, we won't be doing so much of that this term.
We'll certainly be seeing the cardinal and ordinal numbers.
OK.
Later on.
Few decades later, set theory was developed as a foundation of mathematics.
And if you're into the philosophy of set theory, this may very well be what you're interested
in.
But what does it mean to say that set theory can be a foundation for mathematics?
Well, the metaphor is from building.
So when you build a house, before you actually start building the house itself,
you dig down into the ground and set a foundation so that the house won't fall over.
So presumably set theory is supposed to be something secure and solid underground that
we can build the building of mathematics on top of.
Well, how does that work?
Well, the first thing is that in set theory, we can make copies of a lot of mathematical
structures.
So in this lecture course, we'll construct the natural numbers.
So the natural numbers are not basic to mathematics anymore.
They can be defined in terms of something simpler, namely sets.
And once you've got the natural numbers, we can construct the integers, the rationals,
the reals, the complex numbers and all of those. Constructing Z
Q and R from N is not on the syllabus.
I'll say later on in the course in a handout how that's done.
So that means that the universal sets can be treated as if it was a universe of all
mathematics.
So we can represent the natural numbers by a set.
We can represent Q by a set and so on.
And so we can say, okay, when we're doing mathematics, the universe of sets is it,
It's everything.
And we ignore everything else.
OK.
So that's one way in which set theory is a foundation for mathematics is an ontological
way, if you like.
The other question is about provability.
So we might have a complicated mathematical question.
We can sometimes reduce it to a question about set theory.
So.
So we try and prove our theorem.
We might find it.
Our theorem proves itself really easily, using only a very basic set theoretic
assumptions. We might find this is really
sophisticated and fiddly and uses very complicated and powerful set theoretic assumptions.
So we get a measure of how difficult a mathematical statement is or if the set theoretic
axiom we need is a bit unlikely.
How unlikely our statement is.
So.
So there are, we have extra set theoretic axioms.
I've mentioned a few.
The Continuum Hypothesis, Martin's Axiom, the Axiom Constructability,
the Axiom Projective Determinacy - projective determinacy is particularly powerful or,
if you like,
Unlikely.
Okay.
So we get the I mean, so very often in certain areas of mathematics we keep track of which
set theoretic axioms
we need to prove our mathematical results and our list of set theoretic axioms provides
a uniform scale of sophistication.
Anyway, so that's an epistemological way in which a set theory could be a foundation of
mathematics, a way of saying,
yes, how do we know something is true - well we can prove it if we make the following assumptions
Anyway,
so that is set theory as a foundation for mathematics.
And I should say that when I talk about the universe of sets, I'm talking about pure sets.
So sets whose elements are other sets
And so any set that I'm talking about in this course will be if you like, a
bunch of curly brackets arranged in a row.
And there'll be when we dig down through the pairs of curly brackets.
There will be nothing at the bottom.
So I've written an example of a set of a kind that won't appear in this
course because it has elements that aren't sets such as the planet Jupiter.
Okay.
Why do we insist on pure sets?
Well, firstly, it's easier.
So if I tried to define in ordinary life, the set of all oak trees, I've got questions
of vagueness coming up.
So has an acorn that's just sprouting.
Is that an oak tree? Is a dead oak tree an oak tree and so on.
We have we have all sorts of difficult questions that come in when we start trying
to do set theoretic reasoning about ordinary life,
which aren't fair.
in the universe of pure sets.
And the other reason is we're trying to do mathematics for the purposes of this course.
So pure sets turn out to be enough.
So those are what we're going to deal with.
So people who are interested in reasoning about life rather than mathematics won't want
to stick with pure sets,
but then they will have other problems that they will need to get to grips with anyway.
So why study set theory?
Well.
Firstly, because it's useful.
And secondly, because it provides a foundation on which one can build mathematics.