Gödel's Incompleteness (extra footage 1) - Numberphile

The theorem implies that there will be an infinite number of undecidable sentences, true sentences that cannot be proven. So, then, one might hope, well, we could just add those finite numbers as axioms and then we'll have a complete. That is,

incompleteness

refers to the fact that there is this kind of gap between truth and evidence. We would love a complete system where we had a set of axioms and all truths could be proven. So incomplete refers to the fact that you can never complete it. Brady: "How will a mathematician know if what he's working on is really undecidable or falls into Gödel's basket?" It seems like you could have things in your too-hard basket and you'd say, ah, this must "I think that's one of the real challenges for mathematicians.

And I think most of us actually kind of stick our heads in the sand and say, "lalalalala! You know, I don't really want to know about that" because, and I think it's important when you're trying to prove a theorem, that you believe you can prove it. I mean, there are other issues, actually, about provability, which are issues of complexity. We know that there are some proofs that will be of a complexity that our human mind will never be able to navigate. In some ways we feel that number theory yields statements that are easy to write but the complexity, look at Fermat's last theorem, is immense compared to the. claim that there are no solutions to these equations.

And we know, just for mathematical reasons, that there will be claims whose proofs are of a length that even the universe, like, let's consider the universe as a computer, will not be able to figure out before the end of the. time. I think Gödel's revelation changed the conception that people had about mathematics. Because I think there was a feeling, we should be able to prove anything that is true. So I think there was some kind of change. But, you know, that's something we have to deal with. It gives mathematics an interesting kind of complexity that didn't exist before.

So there is something intriguing about the way we as humans can exit a system. And this, in fact, has been used by some to suggest that this is why human consciousness is actually much more than a simple analog computer. Because how can an analog computer get out of the system it's trapped in? However, it seems that we can exit the system to see the meaning of that phrase, enter the system and prove that it cannot be proven. But, you know, someone like Roger Penrose has used Gödel as a kind of challenge to whether human consciousness can ever be captured by something like a conventional computer.

There's a challenge in that kind of idea of ​​saying, well, humans are better than machines. Because the challenge is that even when we work outside the system, we ourselves continue to work within another system, which we assume is consistent, that it has no contradictions, but we have to make that assumption. So I think this feeling of, well, we're better than the machine, well, we have to remember that we're also limiting ourselves within our own logical thinking system. There is a beautiful phrase. Since we cannot prove it, one of Gödel's conclusions is that we cannot prove that mathematics does not contain contradictions.

And that is also very disturbing. I mean, it's amazing. We have been doing mathematics for several thousand years and no one has thought of any contradiction. So that's good evidence that it does work. But that doesn't mean there isn't something really subtle that can block it. And I think 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 to paradoxes, which seemed to be things of a fairly mathematical nature. But there's a beautiful quote from 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 he is consistent. Brady: "When you look at what Gödel did, what was it that made his theorem brilliant, or a leap, or something clever? "For example, what, what was the new thing that he came up with that all the people before had not done? Haven't you thought? "What was the brilliance of this?" Gödel's brilliance was the idea of ​​allowing mathematics to speak about itself. And at the core is the idea of ​​being able to code each statement in mathematics with its own unique code number. And I mean, I like it because one of my obsessions, as people will know, is prime numbers, and Gödel uses prime numbers very cleverly to produce this encoding.

So essentially each logical symbol has its own type of prime number, and the number of times it is used, where it is used, is encoded in the power of that prime. And I think this was, really, an

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ordinary revelation. That mathematics could be self-referential. How could we talk about mathematically proving things in mathematics? But using this coding, there was a way to do it. For me, that was the brilliance of Gödel. Brady: "All statements, all symbols, everything can be encoded using these numbers. "Can you give me an idea of ​​how big these numbers are? It's not like a plus sign is a 7, "and a division sign is a 13, right?

These are like mega numbers?" No no no. They start very small. It's not necessary, so the first logical symbol will have the prime 2, the second 3, 5, 7, so, but the point is that when you're actually taking a math statement and looking at its code number, it will be a product of all these prime numbers, the power of the prime number will also indicate something about the structure of that sentence, so once you start multiplying all these prime numbers, you get incredibly huge numbers. But you are correct. The ingredients are actually the first prime numbers. So yes. Brady: "So if I were to go find Andrew Wiles' famous test and boil it down to its most basic essence, "which is still pretty massive, I could, I could, I could spit out a number at the end. , "which could be printed on a sheet of paper." Yes.

It means that the statement of Fermat's Last Theorem will have a code number, that each logical step in that proof will have its own code number, so you can write the proof of Fermat's Last Theorem as a huge number. Brady: "Has the topic changed? Has the math changed? Or is it just this "landmine that's out there that everyone hopes not to step on?" I think we have to have this arrogant belief that what we're working on we can to demonstrate. It's almost part of the structure of being a mathematician, I mean, you know, it's not just about Gödel, it's about saying, well, maybe this is so complex that my brain is We're not going to be able to prove it. which is surprising how much mathematics we can capture with our finite equipment in our heads.

So I don't think it has fundamentally changed the mathematician's mentality too much. We all have to be careful with that. There is a lovely novel by Apostolos Doxiadis called Uncle Petros and the. Goldbach's conjecture and it's about a Greek mathematician who has been working on the Goldbach conjecture and suddenly discovers that it is set in the 1930s. He suddenly comes across this work by Gödel and it completely undermines his work. And if!? What if this is a true statement that has no proof? And I tell you, Goldbach has a kind of feel for that. Because it's kind of a combination of two things that probably shouldn't have anything to do with each other: addition and the atoms of multiplication, the primes.

So it could be something that turns out to be true, but doesn't actually have a good proof of the axioms. Gödel's

incompleteness

theorem really captured the public imagination because it seemed to show limitations of knowledge, and people like that idea. And it seemed to show that mathematics was not as all-powerful as people thought. But I think you have to be careful here. Because the strange thing is that we can prove that statement to be true, it just works outside the system. I mean, we're still pretty powerful mathematicians. But I think it does show that within any system there will be limitations.

So I think I've spent the last three years on this kind of journey, inspired by Gödel, to look at the other sciences and see if they have their own statements that, by their very nature, may be unknowable. I think there's a kind of feeling, maybe science can know everything, but then are there other sciences that have similar limitations on what they could know? Many things we don't know now, but perhaps there are questions that, by their nature, we can never know. There is a nice story that Hilbert is going to become an honorary citizen of his hometown, Königsberg.

He has been awarded this great honor and he makes the statement from it: Wir müssen werden wir werden werden My German is not good enough. We must know that we will know. This belief that there is no such thing as what he called ignorabimus. Without ignorance. What he didn't know is that Kurt Gödel, in the same city of Königsberg, a few days before, had given his talk on this great new theorem, the incompleteness theorem which shows that the ignorabimus is actually part of mathematics. Brady: "And it also seems like that should have been Hilbert's number one.

It seems so fundamental 'should have been at the top of his list.' Well, it's interesting. Because Hilbert's first problem relates to something that Gödel There was also What I'm interested in is the nature of the infinite. 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 everything real. Numbers? Maybe there is an infinite set between them. Now, here is an interesting example of a challenge, mathematically, we should surely be able to solve it. Thanks to Gödel and Cohen too, you can choose your number. mathematics cannot be shown that this is true or that its negation is so true that either can be included as an axiom, and if mathematics was consistent before, Gödel still is.

Some other interesting contributions, not just to. mathematics, but physics. He was a great friend of Einstein, at Princeton, they used to walk together to the Institute for Advanced Study in the morning. And he looked at Einstein's equations for general relativity and showed that there is a solution to those equations where time is circular. So these loops occur. Now, we assume that physically can't happen because it would involve certain paradoxes like the grandfather paradox, you would be able to go back and kill your grandfather. But it is fascinating that Gödel, once again, was able to demonstrate these slightly paradoxical solutions to the theory of general relativity.

The other intriguing thing he did was that he took the American constitution and discovered a logical inconsistency in it, which completely invalidated any statement you made. And then when he became an American citizen, I think he was going to bring this up and say, well, actually you realize that there's a logical inconsistency that completely invalidates any statement that's here. And I think they encouraged him not to mention that in the sort of ceremony for him to become a citizen. Gödel had a really tragic end, because he became very paranoid when he was in the United States because people were trying to poison him.

And he essentially starved himself to death because he was so terrified that any food would actually kill him. So it's kind of a sad ending to an

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ordinary life. ...a true value. But then when I went to university, I realized that in mathematics you can't have those things. However, when I took this course on mathematical logic and we learned about Gödel's incompleteness theorem,