There we go.

Sorry?

We can do that.

I feel like this one was easy, right?

Very good.

Okay.

So, which of the following are axioms of CF?

Yes?

Ah, okay.

Okay.

Fair point.

That's also a typo.

So it should say axiom of union, but either is fine.

I wasn't drilling down on that level, right?

There's also always like, do you call it the power set axiom or do you call it the axiom

of power sets and like infinity axiom or axiom of infinity or axiom of natural numbers,

you probably also call it, right?

The names don't matter as much as what they say and like the axiom of unions obviously

introduces unions, so you might want to realize.

But yeah, fair enough.

If you just go literally by name as it was in the slides, then this is confusing.

Yes?

Yeah.

Okay, but like...

Yeah, that's fair.

And if it's about these kinds of things, feel free to just ask.

I don't mind answering these kinds of questions.

Yeah.

Apart from that, the rest is clear, right?

I'm assuming.

Okay.

What about the other questions?

Anything else that was confusing?

Oh, and made a paragraph symbol out of the S, interesting.

Axiom of union is correct, axiom of infinity is correct and you need...

Because union tells you you can form the union of all sets in a set, so you have to be able

to form the set that contains all of these sets and for that you need replacement.

Because then you can just take omega to replacement to get the sets that contains all of the Si

and then you can take the union of that.

So you need omega to show that this sequence of sets even exists, replacement to put them

all in a set and then union to form the union of that.

Okay?

Yeah.

Next one.

Okay, so the definition of an ordinal is that it's a transitive set, it's well ordered,

blah, blah, blah, right.

So one consequence of that is every ordinal contains all of its predecessors, so every

element of an ordinal is itself an ordinal.

Yeah, so in particular any x with x in alpha is by definition an ordinal because that means

it's a predecessor of alpha.