Gödel's Incompleteness Theorem - Numberphile

[Marcus du Sautoy] I've been quite obsessed with Gödel's incompleteness theorem for many years because it kind of places this extraordinary

limitation on what we might be able to know in mathematics. In fact, it's quite an unnerving theorem

because at its heart it says there might be

conjectures out there about numbers, for example something like Goldbach's conjecture, that might actually be true

So it might be true that every even number is the sum of two primes

but maybe within the

axiomatic system we have for mathematics, there isn't a proof of that.

The real worry is what if there's a true statement that I'm working away on which actually doesn't

have a proof.

Now his is a big kind of revelation for mathematics because I think ever since the ancient Greeks

we believed that any true statement about mathematics will have a proof. It might be quite difficult to find like

Fermat's last Theorem took 350 years to, before my colleague in Oxford Andrew Wiles found the proof.

But I think we all have this kind of feeling like well surely every true statement has a proof

but Gödel shows that actually there's a gap between

truth and

proof.

I wrote it down here because it's quite cute

So it's one of these cards: "the statement on the other side of this card is false".

So let's suppose that's true. So it means that the statement on the other side of the card is false

So we turn it over and then it says: "the statement on the other side of this card is true".

Well, that's meant to be false. So it means the one on the other side is also false

Oh, but we suppose that that was true, so that's false

So the other side is true, which means that --and you get into this kind of infinite loop.

Verbal paradoxes are fine because you don't expect every

verbal sentence to have a truth value to it.

But then, when I went up to university, I realized that [in] mathematics you can't have those; yet when I took this course

on mathematical logic, and we learned about Gödel's incompleteness theorem, he used this kind of

self-referential statement to really undermine our

Belief that all true statements could be proved.

There was a feeling like we should be able to prove that mathematics is something called consistent.

That mathematics won't give rise to contradictions.

This have been kind of inspired by certain kind of little paradoxes that

people like Bertrand Russell had come up with.

People might have come across this idea of "the set of all sets that don't contain themselves as members"

and then you-- the challenges well is that set in this set or not?

Actually, a really nice kind of version of this is another sort of mathematical