The theorem implies that there will be an infinite number of undecidable sentences,
sentences which are true which cannot be proved.
So, because then, you might hope, well, we could just add those finite number as axioms
and then we will have a complete.
This is, the incompleteness is referring to the fact that there's this kind of gap between truth and proof.
We'd love a complete system where we've got a set of axioms, and all truths can be proved.
So incompleteness refers to the fact that you can never kind of
complete it.
Brady: "How will a mathematician ever know if what they're working on is truly undecidable, or falls into the Gödel basket?
"It seems like you could have things in your too hard basket, and you would say, ah, this must be a Gödel problem."
I think that's one of the real challenges for mathematicians.
And I think most of us, actually, kind of, slightly stick our head in the sand, and go, "lalalalala! you know, I don't really want to know about that" because,
and I think it's important when you're going in to try to prove a theorem, that you believe you can prove it.
I mean, there are other issues, actually, about provability which are complexity issues.
We know that there are some proofs that will be of a complexity that our human mind
will never be able to navigate.
We sort of feel like number theory throws up statements which are easy to write down,
but the complexity, look at Fermat's last theorem,
that, the complexity of that is immense compared to the statement of
there being no solutions to these equations.
And we know, just for mathematical reasons, there will be statements which, whose proofs are of a length that
even the universe, as a, consider the universe as a computer,
it will not be able to crunch out before time ends.
I think that Gödel's revelation did change people's kind of conception about mathematics.
Because I think there was a feeling, like, we should be able to prove anything that's true.
So I think there was a kind of shift.
But, you know, that's just something we have to, have to deal with.
It gives mathematics an interesting sort of complexity to it that wasn't there before.
So, and there's something intriguing about the way that we, as humans, can pull ourselves outside a system.
And this, actually, has been used by some to suggest that this is why human consciousness
is actually much more than just an analog computer.
Because how can an analog computer pull itself out of the system that it's stuck within?
Yet we seem to be able to shift from outside the system to see meaning
of that sentence, go inside the system, prove that it can't be proved.
But, you know, so someone like Roger Penrose has used Gödel as a kind of challenge to whether
human consciousness can ever be captured by something like a, like a conventional computer.
There is a challenge to that kind of idea of saying, well, humans are better than machines.
Because the challenge is that even when we're working outside the system, we are ourselves still working within another system,
which we're assuming is consistent, doesn't have contradictions, but we're having to make that assumption.
So I think this feeling like, well, we're better than the machine, well, we have to remember that
we're also limiting ourselves within our own system of logical thought.
There's a lovely sentence. Because we can't prove, one of Gödel's conclusions is that
we cannot prove that mathematics does not have contradictions in it.
And that's also very unsettling.
I mean, it's amazing. We've been doing mathematics for several thousand years,
and nobody's come up with any contradictions.
So that's a good evidence that it does work.
But that's not to say that there might not be some really subtle thing which will crash it.
And I think that we really had to think about this very carefully
when Russell was coming up with paradoxes about set theory.
We had to have a new conception about sets
because of his challenge of paradoxes which seemed to be things which were quite mathematical in nature.
But there's a lovely quote by one of my heroes, André Weil, a French mathematician, number theorist, who says, you know,
God exists because mathematics is consistent. The devil exists because we cannot prove that it is consistent.
Brady: "When you look at what Gödel did, what was it about his theorem that was brilliant, or a leap or clever?
"Like, what, what was the new thing he came up with that all the, all the people before hadn't thought of?
"What was the brilliance of it?"
Gödel's brilliance was this idea of allowing mathematics to talk about itself.
And at its heart is this idea of being able to code every statement in mathematics
with its own unique code number.
And, I mean, I like it because one of my obsessions, as people might know, is prime numbers,
and Gödel uses the primes very cleverly to produce this coding.
So, essentially, every logical symbol gets its own sort of prime number, and the number of times it's used,
where it's used, is kind of coded in the power of that prime.
And I think this was, really, the extraordinary revelation.
That mathematics could be self-referential.
How could you talk about proving things in mathematics mathematically?
But using this coding, there was a way to do it.
So for me, that was the brilliance of Gödel.
Brady: "All statements, all symbols, everything can be coded using these numbers.
"Can you give me some idea how big these numbers are? It's not like, a plus sign is a 7,
"and a divide sign is a 13, are we, these are like mega-numbers?"
No, no, no. They start really small. You don't need to, so the first logical symbol
will have the prime 2, the second one 3, 5, 7, so,
but the point is that when you're actually taking a statement of mathematics
and looking at its code number, that will be a product of all of these primes,
the power of the prime will be also indicating something about the structure of that sentence,
so once you start multiplying all of these primes together, you get incredibly huge numbers.
But you're right. The ingredients are really the first few prime numbers. So yeah.
Brady: "So if I was to go and get Andrew Wiles' famous proof, and strip it back to its barest of bare bones,
"which is still pretty massive, I could, it would, I could spit out a number at the end,
"that could be printed out on a piece of paper."
Yes. It means that the statement of Fermat's last theorem itself will have a code number,
that every single logical step in that proof will have its own code number,
and so you can write down the proof of Fermat's last theorem as one whacking great number.
Brady: "Has it changed the subject? Has it changed mathematics? Or is it just this
"land mine that's out there that you all hope you're not stepping on?"
I think that we kind of have to have this arrogant belief that the thing we're working on we can prove.
It's almost part of the makeup of being a mathematician is that, I mean, you know,
it's not just Gödel, it's about saying, well, maybe this is so complex my brain isn't going to manage to prove it.
I think it's amazing how much mathematics we are able to capture with our, the finite equipment in our heads.
So I don't think it really has changed too fundamentally the mindset of the mathematician.
But we all have to be wary of that.
There's a lovely novel by Apostolos Doxiadis called Uncle Petros & The Goldbach Conjecture.
And this is about a Greek mathematician who's been working on the Goldbach Conjecture
and he then suddenly discovers, it's set in the 1930s, he suddenly comes across this work of Gödel,
and it completely undermines his work.
What if!? What if this is a statement which is true which doesn't have a proof?
And I tell you, Goldbach has a kind of feel of that. Because it's sort of combining two things
that probably shouldn't have anything to do with each other: addition and
the atoms of multiplication, the primes. So it might just be something which happens to be true,
but doesn't really have a good proof from the axioms.
Gödel's incompleteness theorem really captured the public imagination
because it sort of seemed to show a limitations of knowledge,
and people kind of like that idea.
And it seemed to show that mathematics wasn't as all-powerful as people thought.
But I think you have to be cautious here. Because the weird thing is that we can prove that
that statement is true, it's just working outside the system.
It's, so, you know, we're still pretty powerful, mathematicians.
But I think it does show that within any system there will be limitations.
So I think, you know, I spent the last three years on this kind of journey, inspired by Gödel,
to look at the other sciences to see whether they have their own statements
which, by their very nature, may be unknowable.
I think there's a kind of feeling, maybe science can know it all,
but, so are there any kind of other sciences which have similar, sort of, limitations
on what they could possibly know?
Lots of things we don't know now, but maybe there are questions that by their nature, we can never know.
There's a lovely story that Hilbert goes to become an honorary citizen of his home town of Königsberg.
He's given this great honor, and he makes his declaration:
Wir müssen werden wir werden werden
My German isn't good enough. We must know we shall know.
This belief that there is no what he called ignorabimus.
No ignorance.
What he didn't realize it that Kurt Gödel, in the same town of Königsberg, a few days earlier,
had given his talk about this great new theorem, the incompleteness theorem
showing that ignorabimus is actually part of mathematics.
Brady: "And also it seems like that should have been Hilbert's number one. It seems so fundamental
"it should have been at the top of his list."
Well, it's interesting. Because Hilbert's first problem does relate to something that Gödel was also interested in,
which is the nature of infinity. So, it's something called the continuum hypothesis, which asks,
is there an infinite set between the countable numbers and the continuum, the set,
the size of the continuum, the size of all the real numbers?
Maybe there's an infinite set between those.
Now here's an interesting example of a challenge,
mathematically, surely we should be able to work that out.
It turned out, thanks to Gödel, and Cohen, as well, that you can choose your answer.
So you can't prove within mathematics that either this is true, or its negation is true
so much so that you can actually put either in as an axiom,
and if mathematics was consistent before, it's still consistent.
Gödel did have make some other interesting contributions, not just to mathematics, to physics.
He took, he was a great friend of Einstein, in Princeton, they used to walk to the Institute for Advanced Study
together in the morning.
And he looked at Einstein's equations for general relativity
and he showed that there's a solution of those equations where time is circular.
So you get these loops happening. Now, we presume that that can't physically happen
because that would imply certain paradoxes like the grandfather paradox,
you'd be able to go back and kill your grandfather.
But it's fascinating that Gödel, again, was able to prove these slightly paradoxical solutions
to the theory of general relativity.
The other intriguing thing he did was he took the American constitution and he discovered a logical inconsistency in that,
which completely invalidated any statement that you would make.
And so, when he became an American citizen, I think he was going to raise this,
and say, well, actually you realize that there's a logical inconsistency which completely invalidates any statement that's here.
And I think he was encouraged not to bring that up at his kind of ceremony to become a citizen.
Gödel had a really tragic end, because he became very paranoid when he was in America
that people were trying to poison him.
And he essentially starved himself to death because he was so terrified that
any food was, was actually gonna kill him.
So it's a kind of sad ending to an extraordinary life.
...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,