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With these foundations we can actually proceed to an important topic.

That's 1.6, classification of sets.

Now this is an important exercise.

Every time you define a notion in mathematics, you immediately ask what is the classification.

How many different objects of this type do I have?

How do I get an overview over this structure?

It's fine to write down a definition, it's fine to write down in this case nine axioms,

but we would like to have an overview. What type of, in this case, sets are there?

But this is something that doesn't only happen in set theory, it's a recurrent theme in mathematics.

So let me write this down.

A recurrent theme in mathematics is the study, well study is very general, let's be more precise,

is the classification of spaces, and I'll explain in a second what a space is supposed to mean,

is the classification of spaces by means of structure preserving maps between those spaces.

So we have to clarify what we mean by spaces, and we've got to clarify what we mean by maps.

A space is usually meant to be some set together, or let's say equipped with some structure.

And in the beginning of the course I told you that we'll put layers of structures on top of each other,

so we'll equip sets with more and more and more structure, and that's the structure I'm talking about here.

So it will be topology as the next step, then it will be a differentiable structure, and so on.

But it could also be other structures like a group structure, so you give an operation on a set,

and so you add structure like this, and every time this principle applies,

the space is usually meant to be some set equipped with some structure, and now you want to classify such a space.

Well, the case of sets is of course a trivial instance of this, because it's a set without additional structure.

Fine, but it's nevertheless a set, and a structure preserving map in the case of set theory simply becomes a map,

because there's no structure that could be preserved.

Well, it's the structure of the set itself, but it's kind of the lowest level structure, and we'll investigate this.

So this is the set has structure, so this is this.

So obviously we need a notion of map definition.

A map phi is a relation.

Well, it's a map phi from a set A to B is a relation such that for every A in A,

there exists exactly one B in B such that phi A, B, full stop.

What does it mean, phi of A, B? Well, it means phi of A, B is true.

You remember, a relation of two variables can be true or false, and if A and B are sets,

then a map from A to B is such a special relation such that it exists for every A, exactly one B in B.

Now, our standard notation for this somehow hides the fact that phi is a relation that eats two variables

and produces a truth value.

The standard notation for this is to write phi A, B, and then to write A to the value A in A,

we assign the value phi of B in B.

And you see, if you say phi is a relation, then this is an abuse of notation because the relation needs to eat two objects

and produce a truth value. But once we agree that for every A in A, there exists exactly one B in B,

such that phi of A, B, then phi of A is precisely this B.

So this is then defined as precisely the B that's unique such that phi of A, B is true.

And that's the standard notation, right? But structurally speaking, well, I don't know whether it's important,

but it's at least structurally correct to keep in mind that a map is actually a relation.

If you don't look at a map as a relation, then you get into trouble because you have to say,

well, a map is such that to every A, there is assigned a unique B.

The question is, what do I mean by there is assigned a certain B?

What does it mean to assign to one element in A, another element in B?

This notion of assigning is not defined. And so it comes from a relation.

So that's the basic idea of a map.