Infinity Paradoxes - Numberphile

Key Takeaways

we're going to talk about four paradoxes that touch on the topic of infinity first one we've got the Hilbert Hotel this is a hotel with infinitely many rooms and at the moment there's somebody in every single room so it's a full hotel and then a new customer rolls up at the reception desk now you might think they got to turn them away because the hotel's full but the manager's clever and here's what he does he shifts the person in room one to room two and the person in room two to room three and the person in room three to room four and so on everyone gets shifted Ford one room and because there's infinitely many rooms you never run out of rooms to put people in and when you've done all of this room one's free so the new guest get put puts in room number one so you can fit in a new guest you could fit in two new guests you could fit in 10 new guests and even if an infinite bus of new customers rolls up you can fit them all in I get gu it's that when I first heard the description I thought well that hotel's full you can't fit anybody in it that's how full hotels work but then once you're shown that you can fit as many new people in the hotel as you like you learn something new about Infinity so your intuitions change and the trick is cuz it's an infinite Hotel there is no final room if there were a final room you could count all the rooms and there wouldn't be infinitely many of them I think he's using the idea to say you know Infinity is interesting it's philosophic Ally interesting because you might have thought things went like this but they don't and we can show mathematically that they don't Paradox number two this is the Paradox of Gabriel's trumpet so the idea is we have this mathematical shape this mathematical object shap like a trumpet it'll start off here and it'll taper and get thinner and thinner and thinner and thinner and thinner and it tends off to Infinity Getting Thinner and thinner all the way so that mathematical object has an infinite surface area but the volume it encloses the air in the center of the trumpet is only a finite volume suppose you were told you had to paint the inside of that trumpet well on the one hand you've got to paint an infinitely big surface so you're going to need an infinite bucket of paint but on the other hand you could just take a finite big enough can of paint tip it into the trumpet and let it filter its way down so it looks like you only need a finite amount of paint to paint an infinite surface area that's really really strange if we're thinking about real physical paint you just can't paint the whole surface because it gets so thin at one end that you're just not going to fit the paint down there so I guess the Clash arises when we think about the difference between the mathematical idea of a surface and the physical idea of paint which is real thick stuff if we had mathematical paint where the molecules had no size whatsoever looks like you need an infinite amount of that to cover that infinite surface area he thought there was something really strange going on here he gave lots of proofs of this idea because I think at first he thought he' made a mistake and lots of mathematicians thought that there was something wrong with the idea of infinity that maybe somehow we should banish Infinity because puzzles like this Clash they show there's something wrong with the idea so we should somehow ban it from our mathematics but in fact Infinity works just fine in mathematics but we have to sometimes change our ideas about how the world Works to fit it in number three so this is the puzzle of the dartboard so we suppose we've got a dartboard and we've got a doart and we're going to throw the doart at the dartboard and let's just suppose we're guaranteed 100% chance to hit the dartboard and then we ask the question think about the exact center point of the do and we think about an exact point a mathematical point on the dart board and we ask the question what's the chance our Dart is going to hit that exact point and it turns out we can't give a sensible answer so suppose on the one hand that we say it's got a chance greater than zero of hitting that point well there's a problem with that if it's got a greater than zero chance of hitting that point the same goes for every other point on the dart board and there's infinitely many of them in fact there's a big Infinity of those mathematical points when we add all those chances together to give us the chance that the dart's going to hit the dart board at all we end up with an infinite probability we can't have an infinite probability you can't get a probability bigger than one that something will happen Okay so we got a problem with the idea that the chance of the dot hitting that point is bigger than zero so on the other hand what happens if the chance is zero but that's also strange because if the chance of hitting that point is zero we could say the same for every every point so the chance of hitting any point on the dart board is zero it's not going to happen but that's really strange because we're sure the dart's going to hit the dart board somewhere again one thing philosophers say about this is the idea of banning Infinity so we say don't think about the exact mathematical point of the dart we think about an area any real physical Dart has an area at its Center and we think about what's the chance that it hits an area on the dart board and and then we divide these up in a finite way and the problem goes away as the area gets bigger the probability that the dart hits there is bigger as it gets smaller the probability the dart hits there gets smaller so it's this kind of granular nature of our existance that gets us out of trouble yeah that's right so if everything that we think about is an area rather than an exact point the problem goes away but that's not completely satisfying after all we can think of an area getting smaller and smaller smaller and smaller and tending to Infinity shouldn't we have a decent answer to this problem when we get to the infinite case and you know maybe we don't have a very good answer to it okay so here's the fourth Paradox this is perhaps the most interesting one of them all because it relates to what rational humans should do in betting games you go to a casino and they put a pound in the pot and they say we're going to flip a Fair coin and when it lands on a tail you get whatever's in the pot if we flip ahead the casino doubles what's in the the pot and we play again if it gets a tail you get what's in the pot if it's a head they double the pot and we carry on playing and then they ask you the question how much would you pay to get into this game name your price you've got to pay something to play what would you pay and I think most people think about this a bit and say something like maybe a few pounds maybe if you're uh Rich you'd say 20 the mathematical Theory on the other hand tells you bet whatever you've got stake any amount of money you can get your hands on on this game because the expected winnings is infinite your expected take-home is infinite if you play the game so the way we work out the expected take-home winnings is we look at each case if you flip a tail on the first go how much would you win added to if you flip a tail on the second go how much would you win added to if you flip a tail on the third third go how much would you win and we sum all of these but because the game hasn't got a fixed end point we sum infinitely many of these and in fact what you would the expected value of each play of the game is a half so we're adding a half to a half to a half to a half infinitely many times that's why the expected value of what you'll take home is infinitely big but we wouldn't bet our house would we we wouldn't bet everything I wouldn't and I don't think you would and what does that say I think it says something interesting about perhaps about us um perhaps it says that we're more risk averse than the mathematical Theory would have us believe um perhaps it says that the value of money changes the more you acrw perhaps it says something else that we haven't quite worked yet that we should figure into what makes us rational reasoners that we haven't yet taken account of in the maths you're sort of saying that as if humans are wrong to not bet everything on it but surely if humans did that casinos would set that game up and own all their houses by now well good point to make the maths come out the casino has to guarantee so you have to know in advance that they will bet as much as they need so there would have to be a casino with an infinite amount of money potentially to put into this game so one way out of it is to say look there's no casino with that amount of money and if we say after X amount of money has been put in the pot that's it that's all the casino can do then the numbers change and it turns out you would be irrational to put lots of money into it we have this mathematical theory of rational decision- making and most mathematical theories don't crumble when you put Infinity in they work just fine so why is it that theories about us how we should or shouldn't behave go really weird when you put Infinity in so suppose we take all the natural numbers together none of those are infinitely big each one is a finite number but there's infinitely many of them so if we were asked to count them we'd say there's an infinite connection