Okay. I mean, to be fair, if it's not showing an image, it might as well just shut off entirely,

right? Okay. So we were talking about axiomatic set theory. And we know that naive set theory

doesn't work because Russell's paradox, we've talked about that before. So the problem here

is ultimately the comprehension axiom because the comprehension axiom basically says that for every

property there is a set and the property is just a first order proposition, which basically means

the comprehension axiom makes sure that sets are basically just first order propositions. So

technically you're having, if you do naive set theory, you have a first order theory that allows

you to talk about first order propositions and that has some kind of self-referential problematic

stuff built in from the get-go. So what do we do instead? Well, if we can't have full unrestricted

comprehension, then we're just going to weaken it and we're going to weaken it by putting some

guard in here. Okay. So instead of saying for every property there is a set, we're going to say for

every property and every set there is the subset of all the elements of the set that satisfy the

property. Okay. So given a set S and a property phi, I can now form the set of all X and S that

satisfy property phi. That's perfectly fine. And that way I don't get Russell's paradox as a like

proof that something is inconsistent. Instead I get a proof that there is no set that contains all

sets. Why? Because if there is a set that contains all sets, then from that set I can filter out

those that don't contain themselves and then I get Russell's paradox again. So I get the exact

same contradiction, but now it's not a contradiction in my set of axioms. Now it's a contradiction

which disproves the assumption that there is a set that contains all sets. Yes.

Yes. Yes. Quantification over subsets. You can't do that in first order logic. You can't do it in

first order logic if you want to quantify over subsets of your domain. Right. So if say we have

something like piano arithmetic, right, and then I can't do something like for all S subset of

omega, right, that doesn't work. There is also no other way to make that work because the only way

that I can specify a subset of the domain is if I have a property, for example, and I can only have

a property in first order logic if I already have like a relational constant in my signature and then

the corresponding axioms. And so for example, I can't do something like for all prime numbers.

Well, what I can do instead is either I can extend my signature by a new symbol prime and then say

something like for all x, prime x implies whatever. But now I have to like have added it to my signature

already, which I'm only allowed to do in a conservative manner if I can define what a prime

number is. And of course, like for prime numbers, I can actually define that. What I cannot do is

quantify over arbitrary subsets, which is even worse, right. So I can't say something like for

all subsets of the natural numbers, that or this or that, or for every finite subset of the natural

numbers, there's the sum of all the elements of that, right. These kinds of things I just can't say

in first order logic, yes. In naive set theory, I can quantify over subsets because like the elements

of my domain are sets and the subsets are also elements of my domain. So if I can quantify over

all sets, in particular, I can quantify over subsets. And of course, therefore, by any other

axiomatic set theory, this will allow us to quantify over subsets, which is exactly why we

need something like an axiomatic set theory if we want to have a proper foundation for mathematics,

because we need to be able to quantify over subsets and these kinds of things. Okay. Good.

Are we happy with this part? Yes. Yes. The axiomatic separation tells you that if you

already have a set, then from that set, you're allowed to filter all elements that satisfy a

particular set. Yes, exactly. Once you have a set, you can filter out by any property. So in

particular, if you have any set S, you can immediately define the empty set again, because

you can just say the empty set is the set of all X and S such that something contradictory.

Sorry? With the empty set itself, you can't do anything because you can only like, this allows

you to exactly define subsets of some set that you already have. And of course, with the empty set,

all subsets are just the empty set. So you don't get anywhere. Sorry?

It's not entirely related. The reason why we don't get a paradox is because we can't form the set of

all sets that don't contain themselves. The only way we can form the set of all sets that don't

contain themselves is if we have some superset of the set of all sets that don't contain themselves,