In this video, I'm going to talk about the first few axioms of set theory so that they
they can all be phrased in an appropriate,
a formal language, the formal language is first-order predicate calculus
with a binary function
symbol to refer to 'is an element of'. If you are not that fluent in logic.
The details of language don't matter that much for this course.
So it won't matter for the theorems in this course whether we're using first order, logical
second order logic, for example.
And actually I'll just use mathematical English a great deal.
So I say more about that in the first hand out. The axioms.
all of the axioms, by the way, are listed in handouts two and three, informally in
handout two and formally in handout three.
So the handouts are extra text that you can refer to if you want to.
And they will be on the course materials section of the website.
OK.
First, the axiom of extensionality.
I should say that it isn't at all obvious that the axiom of extensionality should be
regarded as true.
Two sets are equal if and only if they have the same elements.
In ordinary life, this is weird.
So in ordinary life, we might say, we would end up saying that the set of vampires
is equal to the set of mermaids.
Because there aren't any of either.
This seems weird.
If you're a philosopher, you may have heard of intentionality.
And the reason, perhaps a reason we regard saying that the set of vampires is
equal to the set of mermaids is that we can imagine a world in which that isn't true.
So we can imagine a world in which there are vampires, there are mermaids, and they're
not the same thing.
Okay, so what the axiom of extensionality is, is really the definition of equality for
sets, and for pure sets.
It works.
And so we're going to say that two sets are equal if and only if they have the same elements.
Okay.
The empty set axiom. We don't need this. From the axioms of first order logic and the other
axioms we can actually deduce it.
But, whatever.
We will say it because it flags it up as important. The empty set exists, and therefore in
particular there exists some sets.
Okay, next, the axiom of pairs.
This is the first of a whole series of axioms that allow us to build more complicated sets
out of simpler ones.
Okay.
The axiom of unions sort of goes in the other direction.
If A is a set, then the union of all the elements of A is a set.
So if you hand me a box full of a bunch of parcels, I can unwrap all the parcels and
put the contents in a box.
So.
So the union of all the elements of A
There is a sense in which it's one level lower in complexity than A itself.
The subset axiom scheme.
So this isn't one axiom, it's an infinite set of them.
If capital A is a set then set all elements of A satisfying a property phi, that's a set as well.
Okay.
Okay, this this this set of axioms.
We can already construct a load of different things.
So I talk about some of those in the lecture notes.
I'm just picking out a few examples from for for this video.
The set whose only element is the empty set exists by the axiom of pairs, because when
I stated the axiom of pairs, let's have another look at it.
That's not it.
This is. When I stated the action of pairs.
I didn't say that A and B should be different.
They can be the same.
So the set whose only element is the empty set is a set, and is different from the empty set
itself.
Why?
Because it has a single element and therefore isn't empty.
Okay.
So I can carry on putting pairs of curly brackets around and end up with a
sequence of sets where it's possible to prove that they're all different.
So you've got infinitely many different sets.
And we can also define union of two sets.
The set difference, We can define intersection.
The union of A and B is you apply the axiom of pairs to A and B to get the set whose elements
A and B take the union of all the elements of that.
And we get A union B and the other two sets can be defined as subsets of A union B.
Okay.
So we can do a lot of stuff with the axioms we've got already.
There are things that we can't do yet.
We haven't yet shown how to construct any infinite sets, for example.
But we'll be coming to that in the future.