Name: Incompleteness Theorem
Uploaded: 2018-02-11T18:28:37.000Z
Duration: 13 min 53 s
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[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

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

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

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

Kind of verbal Paradox is all those numbers that can be defined in less than 20 words so for example

[1729] which is the taxicab number that Ramanujan and hardy talked about you can define that in less than 20 words?

It's the smallest number which is the sum of two cubes in two different ways so that less than 20 words

So then you define the following number then the smallest number which cannot be defined in less than 20 words

Now I think if you count add up, I've just defined that number in less than 20 words, and you go well

That's a bit worrying because surely that is a sort of kind of number

We might define the smallest number which has a certain property.

So there was a real feeling that these paradoxes these set theoretic paradoxes were beginning to be a real challenge

To mathematics at the turn of the 20th century and David Hilbert one of his

big problems that he challenged mathematics with in the 20th -- 20th century

[his] 23 big on unsolved problems the second one was [to] prove that mathematics was in fact consistent and

included in that was that every true statement should be provable, but what a shock

he got actually 30 years later along comes this Austrian logician Kurt Gödel who blows this idea that we can prove math is

consistent out of the water and shows there are true statements which can't be proved within any mathematical system.

How does mathematics work? We set down things we called axioms which are the kind of things we believe are [the] way

numbers, geometry works, so for example if I take six and I add seven to that

I don't think it's going to be different from taking seven and adding six to that. That seems so blindingly obvious

and that's one of the axioms [of] the way numbers work. So maybe somewhere out there that [goes] wrong, but I don't really care

I'm interested in a mathematics where that is true of all numbers.

And I have rules which allow me to make deductions from those axioms

And that's how mathematics works. Each time we make these logical deductions

we expand the conclusions from these axioms. We can add new axioms if we feel like you know what we haven't actually

and that was somehow the -- the mission was we want to have a set of axioms which are strong enough and that we

believe are basic about numbers from which

we can deduce all through statements that we will have a proof and so there was a feeling like well, yeah.

Well, maybe we haven't got all of the axioms and if we got a true statement which can't be proved

We could add that as a new axiom and it will expand what we can prove within mathematics

So this is very important for Gödel because we are trying to prove that there will be a set of axioms from which we can

very clever because he cooked up a way of allowing mathematics to talk about itself.

So what he produced was the thing we now called the Gödel Coding.

So he showed that any statement about numbers

has its own particular code number. In fact you use prime numbers in order to make this coding. So every statement about

mathematics could be turned into a number so for example the axioms of mathematics from which we are trying to make all of our

deductions they would have code numbers and

true statements about mathematics, so for example Fermat's last theorem for example will have a code number, also

false statements like "17 is an even number"

OK so what do these code numbers look like? They're obviously whoppers.

every code number can be worked backwards into a statement -- not every number will have a meaningful mathematical statement

but it's more interesting the other way: every mathematical statement will have a unique number associated with it

A bit like how most things in a computer [have] zeroes and ones. Very good. So if I'm typing away

and I write the name 'Gödel' that be changed into ASCII code. It will have a number

associated with it. So Gödel cooked this up

but why is that useful? Because you can now actually talk about proof in mathematics using these numbers

So you can start to talk about mathematics using mathematics. So for example

you might want to know well, is this particular statement

That -- I'm going to completely simplify this but it'll give you a

feel for it. Essentially it's a bit like saying any statement

whose code number is divisible by the code number of the axioms

will be provable from the axioms. That's an incredible simplification

but it's good because it means now we can talk about proof within --

mathematically to say that something is provable is to say for example that it's code number

So now Gödel challenges you with the following

cannot be proved from the axioms that we have for our system of mathematics"

So this is actually something that has a code number

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Incompleteness Theorem

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