Lex Fridman

Some of these are quite basic, but it would be nice to kind of give people a sense of what it means to be an axiom, like what kind of basic...

facts we can lay on the table on which we can build some beautiful mathematics.

Well, there's also, just to give a flavor, there exists a set with no elements called the empty set.

For any two sets, there's a set that contains exactly those two sets as elements.

For any set, there's a set that contains exactly the elements of the elements of that set, so the union set.

And then there's the power set.

For any set, there's a set whose elements are exactly the subsets of the original set, the power set.

In the axiom of infinity, there exists an infinite set, typically, a set that contains the empty set and is closed under the operation of adding one more element.

Back to our hotel example.

That's right.

And there's more, but it's kind of fascinating.

Put yourself in the mindset of people at the beginning of this, of trying to formalize set theory.

It's fascinating that humans can do that.

And we should say in set theory, consistency means that it is impossible to derive a contradiction from the axioms of the theory.

So it means that there's no contradictions.

That's a consistent axiomatic system that there's no contradictions.

Maybe a quick pause, quick break, quick bathroom break.

You mentioned to me offline, we were talking about Russell's Paradox and that there's a nice, another kind of anthropomorphizable proof of uncountability.

I was wondering if you can lay that out.

Oh yeah, sure, absolutely.