Kurt Godel: The World's Most Incredible Mind (Part 1 of 3)

should say at the outset Kurt girdles perhaps not as famous as many of the big thinkers who I think girdle belongs in this little company such as Darwin Newton Einstein Aristotle and the like main reason I think is logicians and mathematicians don't get their Jew I think you know they don't just it's hard to understand what a mathematician does you can explain to people what was the basic ideas of Darwin or the basic ideas of even Newton Einsteins a little bit harder perhaps but since we have many very good popular science writers who tend to particularly focus on physics a lot of people have at least a good idea of what basics of some of Einstein's work and Newton's work but mathematicians I think get the rough end of the stick sometimes and Kurt gödel certainly belongs in that sort of company I think the idea of the incompleteness theorem which is what I really want to focus on tonight is one of the truly revolutionary ideas of modern thought absolutely monumental result one of the major results in logic many people have said could girdle is right up there with Aristotle and that the incompleteness results were the first major results in logic since Aristotle so this is really big stuff it will involve a little bit of logic and mathematics but I will tread very gently through the technical details in fact why I chose girdle apart from anything else is that the the idea itself is very simple the incompleteness theorem that we'll get to shortly is a very intuitive and simple idea intuitive after after the fact but groundbreaking so here is kurt quick biography born in 1906 in the city of Brune now in czech republic graduated in 1924 studied physics mathematics and philosophy the university of vienna graduated the DPhil in mathematics in 1929 with a spectacular thesis which I won't talk about tonight and continuity's work there the University of Vienna until his departure for the US in 1940 then he took up a position with a good friend he visited the u.s. a couple times prior to that and had met Albert Einstein's very good friends without mine Stein and he ended up at the Princeton Institute for Advanced Study and regular regular walks his daily routine of going for walks and chats with Einstein and he remained there until his retirement in 1976 and died a couple of years later from starvation and exhaustion you might think a sad end to such a great life but if you're a rock star you aspire to die in a plane crash like Lynyrd Skynyrd or Buddy Holly Otis Redding will the second best drown in your own vomit like Jimi Hendrix but that's kind of you're a rock star that's what you're after you're a mathematician and a logician dying of starvation and exhaustion is right up there with drowning in your own vomit you might think that the mathematician and logician the Personality Disorder was redundant that that would be a little bit a little bit mean-spirited so some of these achievements 19:29 his doctoral dissertation on the completeness of first-order logic I won't explain what this means but that was a major result at the time the age of 23 the results will mostly focus on the first and second incompleteness theorems at the age of 25 then did some groundbreaking work in set theory which had actually been carried on through most of his career but perhaps the most famous of these results as a result in 1940 and if we get time I might just mention these novel solutions to Einstein's field equations just because they're so quirky and give you some insight into the diversity of things that could go to carried around in his head he was a foremost magician and mathematician but he actually made major contributions to general relativity as well but in the most quirky in sort of odd way so hopeful have time to just say a little bit about them those solutions to Einstein's field equation so like any big thinker you've got to understand a little bit about the times that they were thinking and no exception here and cook doodle came along a fascinating time for mathematics and logic in an earlier 20th century the foundations of mathematics were in crisis and it's it's hard to overstate how serious things were in mathematics at the time Bertrand Russell had shown that set theory was inconsistent set theory being the foundation or at least that was the hope for everyone that set theory was the the mathematical theory that would underpin all the rest of mathematics is truly rigorous axiomatic system was that it was the hope of mathematics and Bertrand Russell show that it was inconsistent it's not just has a bit of a floor or a bit of a blemish it's that's about as bad as it gets from mathematician on the other side of things can't or had proved that if we had one infinite set such as the natural numbers the counting numbers one two three four five six and so on there are infinitely many of them if we have one infinite set and surely we do because at least we have the natural numbers there are infinitely many infinite sets all there can be ranked in order so it's not just a whole bunch of infinite sets all have the same size but there are infinitely many different infinities hard to wrap your head around what that means without going into the mathematics but this had been shown to be a natural consequence of the standard set theory of the time along with the assumption that there is an infinite set again amazing results but both worrying in their particular ways on the one hand that mathematics is inconsistent just to get a sense of you know who cares mathematics is inconsistent well given that it's inconsistent and given the classical logic that we're supposed to be using here you can prove that two plus two equals five you can prove that I'm Elvis Presley you can prove anything you like the Riemann hypothesis is one of the most outstanding results in mathematics can be proved that in five lines given that you've got an inconsistent theory so it's really bad you can prove all sorts of things which are true but all sorts of things that are not true either in fact you can prove every other contradiction as well one contradiction and it blows up that's worrying for obvious reasons the infinite stuff worrying for more subtle reasons just was not clear that people really had a good grip on what infinity was that there could be infinitely many different infinities one infinities bigger than the first infinity then surely the first one wasn't really infinite you know this must be something bigger than and it can't be Infant so people were worried that they even had a good rip on what infinitely many objects of any kind meant and people were trying to find a consistent set theory and at the same time many of them were weary of Cantor's work on infinity so long combs Frager and he tried to set things on a firm basis he thought well no wonder we don't understand all this business about infinity we don't even understand the number to know what on earth is number two what's the number three what are the natural numbers about what are they we know how to use them and we know how to say things like two plus two equals four but what are they and he worried about this and most other people thought he was just mad you know because we know what number two is what the hell are you talking about but he he really worried he thought this was the root of a great deal of the problems in mathematics and people didn't have a firm understanding of what mathematics was about so he said things on a very firm foundation laid out the assumptions axiom attires basic arithmetic and as often happens once you make it very clear what's going on problems jump out at you and this is what happened to Frager or Bertie Russell who was very influenced by frege's work and had been reading Frager realized that there was an inconsistency in Frager system and he wrote this very nice letter to to Frager a matter of days before his monumental work went to press saying you know did professor Frey go I if I may be so bold that suggests that um there's an inconsistency in your theory and Frager for what he you know a great credit to Frager he said saw it immediately and realized that this was devastating what he didn't realize actually was this problem is generalizable it wasn't just a problem for Fraga it was a problem for mathematics at the time so they get a grip on what was what this contradiction is a very very old philosophical puzzle called the liar paradox which is just simply this sentence is false if if I think about that sentence for a moment two possibilities either it's true or its false right so if it's true then what it says is the case and what it says is that it's false so if it's true it's false that's bad right must be false if it's false well that's what it says it is it says that it's false so if it's false then it's true so if it's true it's false and it's false it's true contradiction well contradiction on the assumption that it takes a truth value at all but why shouldn't it looks like a perfectly well former sentence of English and philosophers you know you might think you know we don't have a lot to do and we worry about things like this disproportionately you know philosophers have worried about this for a couple of thousand years and be quite honest we're no closer to a solution then we were 2000 but better we've had a lot of fun in the meantime Russell saw that this problem was generalizable to mathematics as well so I won't go into the details but the idea is very simple you just consider a set so you can get a set is just a collection of things the set of natural numbers the set of people in this room the set of chairs in this room these are all sets collections of things you have sets of sets so Russell said okay what about the set of sets that are not members of themselves and if you've done a little bit of mathematics or logic you'll see that this is just the liar sentence in set theory form okay so once you form this set then you just ask the question okay is that set a member of itself or is it not if it is then it's not if it's not then it is you get exactly the same conundrum as you do with the liar paradox so the Russell set can be proven to be both a member of itself and not a member of itself that bad mathematics is not supposed to license those sorts of things this is when Russell announced his result Frager it was a problem for Frager but it was also a problem for just a general set theory in a way Frager had just made the problem so transparent because he'd been so clear about his assumptions now shifting to the infinities and Cantor's work just to get a flavor for this it's not very important but it is kind of well I'll admit I'm a nerd and I love this kind of stuff so I just want to share with you some of the the delight in set theory so here's what Cantor is mucking around with an important operation on sets is the power set so the power set once you've got a set of things say I've got the set of people in this room I can then think of the power set of that which is the set of all subsets so it's the set that consists of me and there's a set consists of Meredith and there's the set that consists of the gentleman here in the front row then there's a set that consists of Meredith and me and all the combinations okay all the ways you can pack that up that's the set of all subsets that's called the power set of a set and according to set theory you can always form the power set if you've got a set you can have a power set of it no problem so so here's an example take the set that consists of just two things 0 and 1 so the set that led break those curly brackets the first little curly brackets here just mean that that's the set 0 and 1 considered as an object together one thing the collection of 0 & 1 together that set has two things in it 0 1 the power set of that that has well the empty set is always a subset of a given set it's got the set it contains just a 0 just the 1 and 0 in one so the power set has got 4 things in it 1 2 3 4 cardinality cardinality of a set is in some sense the size of the set how big the set is how many elements in it if you like so if two sets have the same cardinality if phenomenally if the elements of one can be put in one-to-one correspondence with the other okay so what that means that this is actually crucial insight of Kandor was that you don't have to count think

Channel: globalbeehive

Views: 287,159

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Keywords: Kurt Godel, Incompleteness Theorem

Id: i2KP1vWkQ6Y

Length: 14min 59sec (899 seconds)

Published: Mon Oct 17 2011