Name: 哥德爾的不完全性(額外的鏡頭1) - Numberphile(數字愛好者) (Gödel's Incompleteness (extra footage 1) - Numberphile)
Uploaded: 2021-01-14T10:38:04.000Z
Duration: 13 min 24 s
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The theorem implies that there will be an infinite number of undecidable sentences,

sentences which are true which cannot be proved.

轉承詞

So, because then, you might hope, well, we could just add those finite number as axioms

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

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

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.

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.

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?

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

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.

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,

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