Infinite sets can have the same size even when one appears to contain fewer members, as with the natural numbers and the even numbers. The continuum hypothesis concerns different sizes of infinity, and Paul Cohen’s work showed that it is independent of the set-theoretic axioms discussed here.
516 words · auto-generated from the episode video
16:21>> We're moving forward to 1963. Independence of the continuum hypothesis. >> Yes. Um, this is a math one, but I kind of know something about it. Okay, this was mathematician Paul Cohen. He invented the technique of forcing at Stanford University, and he demonstrated that the continuum hypothesis is independent of Zermelo Frankle set theory. The continuum hypothesis has to do with sizes of infinity. For example, like the natural numbers 1 2 3 4 5. Naturally, you can count them, right? You can be like one is the first one, two is the second one, so on and so forth. If I were to ask you um are there more natural numbers than even numbers?
17:03Naively you would say yes because you know even numbers don't have one and three and five. But there's infinite of both. So it's not really a good argument to say that there's more of the natural numbers when both are infinity. In fact, what I can do is I can count all the even numbers. I can say two is the first one, four is the second one, six is the third one, and now I have a one to one mapping. Right? So for every single natural number there's an even number, which means they're really the same size of infinity. It turns out um George Cantor discovered um way back that even rational numbers so fractions you could imagine fractions seems like there's
17:44more fractions than natural numbers cuz you've got even for just the number one I can make 1/2 13 1/4 right I can I can make an infinite number of rational numbers that are just corresponding to the number one but it turns out there's a schema for counting the rational numbers such that every single one has a one to one mapping to the natural numbers. So rational numbers are the same size of infinity as the natural numbers. What about real numbers? Things like pi or e or ah pi squ >> turns out those you cannot count. George cantor proved that as well. So the continuum hypothesis is you've got two sizes of infinity. You've got the size
18:24of the natural numbers which is countable and then you've got the size of the real numbers which is uncountable. meaning there's no way that I could assign like this is the first real number and the second the continuum hypothesis is is there something in between is there a size of infinity that is intermediate >> um the hypothesis is no >> okay >> and he proved this very specific thing saying that it's independent of this certain type of set theory it resolved Hilbert's first problem and earned him the 1966 Fields medal this is a big one I mean America has a lot of Fields medals I picked this one partially because I kind of understand it and partially because in a lot of the lists
19:05this is a big one. >> Uh Hilbert had a whole lot of problems that seem to be getting solved over the course