This might be more of a meta question, but is our mathematics sound and complete?

That's an extremely good question. You would probably want to rephrase this,

but are our mathematical calculei sound and complete? That was a question that worried

mathematicians about a hundred years ago tremendously. They were getting interesting results.

They were getting proofs for a theorem that says, take the unit sphere, think of a tennis

ball, and you can cut that into finitely many pieces. Take it apart, re-assembly, have two

unit spheres of the same size. Wow! Who knew? Whenever you do something miraculous, you

ask yourselves, am I cheating or not? Is this really true? People were having the question,

how is this working? That was when they really developed the formal theory of this.

It turns out that you can develop logics. First of all, logic, which is something we'll

learn after Christmas, is kind of the logic that came out as the best possible one, unless

you're initiated. Let me tell you a little bit more of this story. Around 18 something

or the other, late 19th century, people were getting all these worrying results. They were

looking for something like a calculus for mathematics. A guy called Georg Cantor came

up and said, if we think everything is sets in maths, then we can do wonderful things.

Functions are sets, sets of input-output pairs, and so on. Manifolds are sets, sets of sets

of sets of sets, natural numbers are sets, and those kinds of things. Then a guy called

Russell came, veteran Russell. Did I tell you about the set of all sets that don't

contain themselves? Let me tell you a little bit more about this. Think about sets. There's

the set of all students of AI that are present in Hürzal 10. There's the set of all students

that are on Stuttgart, bigger set. There's the set made of this cup of coffee and you

and, I don't know, my sweater. You can make sets, whatever you want. You can ask yourselves,

are there any very big sets? Well, there's the set of all natural numbers. That's fine.

There's the set of all real numbers and we know that that is bigger. How about the set

of all sets? Well, that's easy. I can make a set out of everything. Now you can ask yourselves,

and that's what you want to ask yourself with sets, is what are the members of the set of

all sets? Well, obviously sets. But this one, actually, is kind of scary. It contains itself.

The set of all sets is a member of the set of all sets because it's a set. Ha! Okay,

so if that is too scary for you, then look at the following set. S being the set of all

m such that m is not an m. I can even write it down. The set of all sets that do not contain

themselves. Now you ask yourself the question, is this set a member of itself? Okay, let's

test. Let's assume S is a member of S. Then, if it's in there, then it's one of these m's.

Right? So, I know m is not an m. So, oh yes! Ah, maybe that was a mistake. Let's try the

other way around. Let's assume that S is not an S. That makes it one of those. So it must

be in there. Just by writing down a set, we've ran into a contradiction. Unsoundness. If

there ever was one, A and not A. There it is. Extremely worrying. Okay? Even more worrying

than these unit spheres. Okay? And then there was this guy Cantor, who was in the 1870s

one of the gods of mathematics and said, everything is a set, but we can't even do sets. He based

everything, all mathematics on sets and sets have problems. Okay? Bad. Then that led to

the question, what are we doing here in math? After all, math is the stuff that makes the

eternal truth. Not like physics, where you can make an experiment and everything is wrong.

Right? Math has the eternal truth. Well, maybe not. Okay, good. So, a guy called David Hilbert,

the god of gods at the time in mathematics said, look, this is not a good situation.

We have to do something about it. And essentially what he said was, let's sit down and make

a sound and complete calculus for mathematics. Where we have this lingua universalis, a language

that can cover all of math, for instance, the language of sets. And it's still sound

and complete. And we would like to actually use math to prove that it is. It's called

Hilbert's program. Hilbert, by the way, has said, oh, and by the way, here's the first

step. There's one. Right? For a tiny piece of math, I can already do this. Ha ha, now

you do better. Okay? And this one happens to be sound and complete. Okay? But you can't