Infinity

so imagine you're Usain Bolt you're like six and a half feet tall you have a couple Olympic medals no biggie you show up to your final Olympic race and your only opponent is a tortoise for some reason he has gold medals around his neck he can talk and he challenges you to a hundred meter race you're confused which you accept his challenge his only condition is that he gets a 50 meter headstart I mean seems fair right he is only a tortoise so you'd line up at the start of the race the gun goes off and the tortoise begins his 50 meter headstart it's gonna take a while so feel free to get a coffee go home take a nap just do whatever you want after a while he finally gets to the 50 meter mark and each Sprint's up to catch up with him but in the time that it took you to get to the 50 meter mark where the tortoise was he's moved forward another 10 meters no problem you just have to catch up to the point to where he is now but wait once you get to that point the tortoise has moved forward another five meters so you have to catch up again this process continues to repeat and repeat an infinite amount of times we can continue cutting the distance between you and the tortoise and half as many times as we want they'll take a finite amount of time to complete but the distances can continue to be cut in half forever let's say it takes you five seconds to run the initial 50 meters then another one second to run the extra 10 meters then another half a second to run the next five all of these times are finite but there's an infinite amount of distances you have to travel so by this logic it should take an infinite amount of time to catch up to the tortoise right you can never catch up to the tortoise and in the end he beats you in the race it makes no sense but this logic says you literally cannot pass him without doing more than an infinite amount of tasks not only can you not beat the tortoise but you can't move anywhere without doing an infinite number of tasks and thus movement from your bed to your refrigerator should take an infinite amount of time so where's our math wrong or is motion impossible [Music] clearly this argument is insane I mean race any tortoise in the world and you'll beat it in a race every single time but where's the flaw in the logic this is known as one of Zeno's paradoxes there's multiple but they all are just the same thing motion as we know it is an illusion the thought is in order to finish the race or really any movement in general you'd first have to get halfway between your starting point in the finish line which in the races case is 50 meters from there you'd have to get to the halfway point between the 50 meter point and the finish line the 75 meter point this continues being cut in half an infinite amount of times this idea can also be reversed and this is where things get interesting if we reverse the sequence what number comes first we can't start at 50 because we can divide that by two we can't start at 25 because we can divide that in half as well in fact we can divide any finite number in half an infinite amount of times this means that there is no first distance to run therefore making motion impossible or maybe not impossible but just an illusion the idea was that because there's an infinite amount of distances being added they must take him infinite amount of time to complete right and not really Xena was wrong to assume that there's an infinite amount of distance to traverse calculus kind of solves the problem with the example of the 100-meter race cutting their distance in half each time just creates what is known as a convergent series when you add all of the infinite amount of terms in this series together you don't get infinity you get 100 a total distance of the race all of the terms when added together converge to 100 and this solves the problem of infinite distance but what about the race versus the tortoise we obviously know that you'd pass the tortoise during the race but how if you run at 10 meters per second while the tortoise only moves at 2 meters per second well after his 50 meter headstart he had passed him in a little over 6 seconds although there isn't mountain of distance to travel there are still an infinite amount of tasks you must perform to pass the tortoise like we just saw the math works out to be just fine but the logic behind it doesn't how can you do an infinite amount of things in a finite amount of time super tasks super tasks provide us with a way of doing infinitely many things in any amount of time you desire its accountably infinite sequence of operations that occur in a finite amount of time countably infinite is really important it'll come in later to explain this will use Thompson's lamp Thompson's lamp is a perfect way to describe super tasks take a Lam lamps can either be on or off nothing else let's say it was your job to turn this lamp on and off and your boss says I need you to turn this lamp on and off at infinite number of times before this hour is up at first you probably will consider quitting but don't this is entirely possible first turn the lamp on did wait 30 minutes and turn the lamp off wait 15 minutes and turn the lamp back on continue doing this waiting half the amount of time from the previous step to do the opposite of what the last step was turning it on if it was off and turning it off if it was on by the time the hour is up you will have turned the lamp on and off a countably infinite number of times but the question is when the clock hits one hour is the lamp on or off it can't be on because when it was on it's immediately turned off in the next step it can't be off because it's immediately turned back on during the next step so what's the final State or is there even one to begin with the super task says nothing about the final state of the object or task in question but only instead of what happens to it during the process still at the end you have completed an infinite number of tasks in a finite amount of time take the sum we'll call it N 1 minus 1 plus 1 minus 1 plus 1 and continue this on an infinite amount of times what is the sum of this if you stop it at an even point it's 0 if you stop it at an odd point it's one but infinite sums don't have a final step just like Thompson's lamp so let's make this a super task that lasts one minute first we have one after half a minute we subtract one after a quarter of a minute we add 1 and continue this until the entire minute is up so what's the answer it could be 1 it could be 0 but it's actually neither if we take our original equation and create a new one called 1 minus n it turns into 1 minus 1 minus 1 plus 1 minus 1 so on when we distribute the negative we have 1 minus 1 plus 1 minus 1 and so on it's our original sum 1 minus N equals n sounds a little bit weird but no problem you can solve this equation easily just add n to the other side and we have 1 equals 2 n divide both sides by 2 and we have our answer 1/2 this is insane and we've added and subtracted an infinite number of ones and we get 1/2 what does this mean for thompson's land when the lamp is on that could be considered 1 when the lamp is off that could be considered 0 what does 1/2 mean for the scenario is it half on half off is it in some other form does the lamp just disappear we don't know it's also assuming that the answer 1/2 is even correct see unlike the 100-meter race where we cut the distances in half each time this is not a convergent series like that was see after we added all the numbers up for the race it converged to a hundred and that's confirmed and that makes sense but with this it's alternating ones and zeroes that don't actually ever converge to an actual value for series like these it's almost dumb to try and attach a value or a sum to it but still the implications for Thompson's lamp stand and they stand unanswered we're coming across problems here we're on the fine line between math and human logic and intuition our math may say one thing while our logic says another is our math wrong or is our few of reality just an illusion in real life you'd beat the tortoise in a race a billion times in a row obviously the Z nerves argument was that there must be a smallest unit of space possible so that there could be an explanation as to how you'd win the race a smallest step that couldn't be cut in half again that would then allow for you to speed past the tortoise and win and it looks like that might be true there's actually a smallest physical and meaningful distance in reality the Planck length 1.16 times 10 to the negative 35 meters any distance smaller than this makes zero physical sense whatsoever the numbers just don't work with current physics the realm of quantum gravity dominates here which is a concept that we don't have an answer for you when trying to solve the problems of quantum gravity or more commonly known as a theory of everything our equations return us infinite answers and in physics infinite answers could not be any more useless but if we attach as small as a distance in the universe our results return finite numbers that we can manage I guess a good question comes up here our math that seems Universal and works everywhere explains space-time but the space-time itself explain our math imagine a box like an old an infinite number of balls you for some reason have an infinite number of balls yourself and want to put them into this box they all have a number on them all of the Naturals so one two three and so on let's say that we have one hour to put all of these balls into the box no problem we'll just make it a super task well put one ball in the start wait 30 minutes and put another ball in wait fifteen more minutes and put another one in and so on clearly after the hour is up we will have a countably infinite number of balls in the box perfect let's do the super task again but differently empty the box and reset your clock okay this time we're gonna put the balls in ten at a time so with a little twist every time we put ten balls in the box will remove that ball step number for example we'll start the clock and put the first ten balls in since this is the first step will remove ball number one after we wait 30 minutes we put ten more balls in balls 11 through 20 and remove ball number two we continue doing this at the same rate as before cutting the time in half each time but still adding 10 and removing one at the same time now if we think about it every time we put balls into the box we add 10 and remove one therefore we have a net gain of nine balls each time right you're correct so at the end of the one hour we should have a countably infinite number of balls again right no because we have a countably infinite number of balls and accountably infinite number of steps there isn't a ball where we don't remove it in a later step ball one is removed on step one ball Tana is removed on step 10 ball one million is removed on step one million through a general term ball n is removed on step N and since we have an infinite amount of tasks to go through in the one hour this means that by the time the hour is up there will be zero balls in the box okay okay maybe we made a mistake let's empty the box again and try it a different way so the same terms apply we have one hour an infinite number of Falls and we put them in ten at a time nothing has changed there now instead of removing the ball that is equal to the step number we're on for example removing ball one at step one ball two at step two and so on we'll remove the highest numbered ball instead so while ten gets removed as step one ball 20 at Step two and so on now we repeat the process of adding balls adding balls one through ten and removing ball ten wait 30 minutes add balls 11 through 20 and remove ball 20 continue this process for the entire hour and then we end up having an infinite number of balls in the box this time this is again insane both of these situations we end up having a net gain of nine balls but depending on which balls we take out and in what order we either have an infinite number of balls or no balls at all both sets of balls have the same values we have a countably infinite number of them at the start we get the order that we put them in and remove them from the box affects the outcome this will be very important later on this goes against our intuition and even now saying this it still doesn't make sense but it is true the idea of infinity infinite sequences of numbers that go on forever end up causing so many problems and kind of make us question everything it's not a number it's an idea infinity isn't a number what numbers are infinitely big it's a definition you can't assign a value to it as weird as it sounds some infinities are literally bigger than other infinities and I'm gonna show you how hey it's me from the future just realize that this is really hard to wrap your mind around and if you have to watch it a couple times to understand don't feel bad about it I'll link the videos in the description that explained these topics pretty well Michael from Vsauce has a really good video explaining this that I highly recommend you watch after this counting is pretty easy you could count to really any number you want if you had all the time in the world for example if you wanted to count to a million and counted at one number per second it'd take you about eleven and a half days it's a little long but manageable however counting to a billion is another task if you were to count at the same rate you did when you counting to a million it would take you over 31 years to count to a billion and a trillion 31 thousand years if you started counting the second you were born you'd only end up being able to count to about 3 billion before you die which is honestly a really small number remember before when I mentioned countably infinite numbers it's an actual thing if you had an infinite amount of time you could count out all of these countably infinite numbers it sounds confusing I know but hear me out the natural numbers 1 2 3 and so on are countably infinite you could go on forever just counting them off but what's weird is that the integers which are all of the natural numbers and their negative counterparts are also countably infinite but logically that makes no sense there should be double the amount of integers as there are Naturals right well not necessarily if you line them up side by side it's what is known as a by ejection this means that you can pair them up one to one both sets have the same cardinality which just means they have the same number of elements or Cardinals Cardinals are just a number denoting a quantity like five circles two rectangles and so on however if we were to label these objects and try and put them in order denoting the order of these numbers are an entire different type of number ordinal numbers to put it simply ordinal numbers order things cardinal numbers represent quantities for example the first ordinal number is exactly that first then you have second third and so on it represents order you wouldn't say you got four place in a race so you got fourth three people finish before you and we can put them in order properly you wouldn't say you have seventh balls in a box you have seven going back to natural and integer numbers it goes against your intuition but with integers you can go off to infinity in both directions and still have the same number of elements to pair up with the naturals it's not the traditional way you'd imagine to list out the integers but it works out perfectly there's also the same number of even numbers and odd numbers as there are both integers and natural numbers they're all the same size using the same pairing method as before it works out perfectly if you were to count forever until the end of the universe 1 2 3 and so on you would finally reach the smallest infinity there is Aleph null Aleph null also isn't a number it just represents the number of counting numbers there are it behaves like one except not really you might think oh I can make a larger number than Aleph null just Aleph null plus 1 right no Aleph null plus 1 is still Aleph nor alpha all divided by 2 is still just Aleph null you could add another Aleph null worth of elements to Aleph null and still the entire set would only have Aleph null elements in it it's like when people say infinity times infinity is just infinity although we can add any number of more elements to our set and still have Aleph null elements the order that we number them in is very important see if we haven't Aleph snow worth of circles and then add another one how do we label this one they're still just Aleph null circles here remember alpha kn'l is a cardinal number which represents the quantity of circles but if we have to label them in order using ordinal numbers what value do we give this one accountably infinite number of circles appeared and then this one did so we need an ordinal number that extends past the finite numbers this circle is labeled with the first transfinite ordinal number Omega up until this point cardinal numbers and ordinal numbers were the same in the land of the finite but here in the infinite they split off into two very different things this circle Omega isn't a bigger number than aleph-null it just comes after it Omega plus 1 comes next Omega plus 2 and so on up until Omega plus Omega or Omega times Omega or mega to the Omega to the Omega Omega it sounds really confusing but there's still only aleph-null elements in this set ordinal numbers just give us a way of labeling them in the order that they appeared because ordinal numbers are different from cardinal numbers after we reach aleph-null they behave in very different ways remember cardinal numbers represent the amount of elements in the set ordinals only placed them in order there aren't more numbers here they're just ordered differently but there are literally infinity is bigger than aleph-null remember how Aleph null is only the set of all natural or integer numbers well what about the numbers between them think about this can you actually really count to the number one so you start at zero and then what comes next 1/2 well it can't be half because you miss fourth it can't be 1/4 because you miss 1/8 it's like the same as the reverse Zeno's paradox what's the first step it'd be zero point and then an infinite number of zeros and then a one at the end but there is no end real numbers are numbers that include all other numbers except imaginary ones real numbers include all decimals all natural numbers all the integers all the irrational numbers like PI and E there are more real numbers between 0 and 1 then there are between the set of naturals and aleph-null remember the pairing method we use before we can do the same thing here just match up the decimals between 0 and 1 to the natural numbers between 0 and aleph-null we can literally match them all up 1 to 1 it still have infinitely many more real numbers left over the numbers from 2 to 3 from 3 to 4 and so on are left without a natural number 2 pair 2 so this infinity of real numbers is literally bigger than aleph-null how much bigger well we'll need to take a look at power sets first a power set just represents the total number of subsets that can be made from a set of elements let me explain let's take the power set for the numbers 3 5 & 7 we need to lay out every single way we can arrange these three numbers well we can have a set of 3 5 & 7 we can have a set of just 3 & 5 we can have a set of just 3 & 7 we can also have a set of just 5 & 7 we can also have sets of just the single numbers themselves so a set of just 3 of just 5 & just 7 finally we can have a set with nothing in it so adding up the numbers of subsets that we can make from the set of 3 5 & 7 we get eight subsets you might be able to see how power setting a set can give you more elements than your original set very fast in fact it's just 2 to the power of however many elements you had in your original set for this example we had three elements 3 5 & 7 2 to the power of 3 is 8 the same number of subsets that we calculated by hand so as we add more and more elements to a set power set of that gets very large very fast for example the power set of a set with the numbers 1 to 100 has 10 to the 30 different subsets that can be made from it that's 10 with 30 zeros after it and that's just with 100 numbers the power set of aleph-null would contain every single set and subset of numbers that there could ever be it's an infinite arrangement of every single way to order Aleph anole things trying to even put in the perspective all the real numbers between 0 & 1 is hard enough but the power set of allen all takes into all the subsets of every number there is this power set is called Aleph one and the ordinal number attached to a low form is called Omega 1 Omega 1 literally represents a number that comes after every single order of numbers that there are listed before it and has to represent a collection of things larger than the subset of all numbers now because Aleph Warren is the next Cardinal number after aleph-null problem shows up there is known as the Continuum Hypothesis what is the power set of the smallest infinity aleph-null the power set of aleph-null things is equal to LF one and Aleph one is surprisingly equal to the number of all real numbers there are or at least we think it can't be less because then there would have to be a set of numbers that encase all of the real numbers numbers beyond our understanding Aleph law represents all the infinity of all natural numbers the integers and so on and Aleph one represents the infinity of all the real numbers which includes every single number there is again not including the imaginary numbers the question is is there an affinity between aleph-null and Aleph one it's a hard question so hard in fact that the only answer we have for it at the moment is that we can't prove it to be false but we also can't prove it to be true that doesn't really help at all does it you can continue going up and up the list of infinite s-- there are larger infinities that mathematicians have come up with that put a lafawnduh shame but to be completely honest I'm not confident enough in my understanding to try and explain those there are infinities so big that it actually hurts me to think about and I think that is one of the most interesting things about humans there are things that we can conjure up that physically make no sense in our universe yet we have still managed to come up with them nonetheless and we can admit that we don't understand our own ideas the difference between math and physics is that we aren't limited to physical constraints and math we have and can literally hypothesize things that cannot be represented in the physical world we live in like I said earlier infinity isn't a thing it's not a place or a number that you can reach it's an idea the biggest of ideas actually and ideas pushes to further understand our place in the universe that we live in and perhaps even learn about things that lie beyond our reach over the course of creating this video I found myself questioning numbers and how they work and how sets like the number of integers are equal to the number of Naturals things that aren't really intuitive but are proven to be true brilliant that org played a big role in improving my understanding of infinity while researching for this video I realized that brilliant had a course on number theory which conveniently had a section on exploring infinity it covers all the topics I covered in this video cardinality counting to infinity and even more paradoxes covering infinity that I didn't have time to cover like Hilbert's hotel if you're interested or maybe even if had been confused on infinity still I highly recommend you check out brilliance course on number theory it actually helped me wrap my mind around the concepts if you head over to the brilliant org slash aperture or click the link at the top of the description you'll be able to sign up and receive 20% off of our premium subscription that will allow you to take as many of brilliance courses as you'd like [Music] [Music] you

Channel: Aperture

Views: 694,473

Rating: 4.9011521 out of 5

Keywords: aperture, aperture infinity, infinity, what is infinity?, aleph null, how to count to infinity, zenos paradox, is motion an illusion, The Sizes of Infinity Explained, sizes of infinity, how big is infinity, infinity paradoxes, supertasks, infinity plus infinity, what comes after infinity, aleph numbers, what is the biggest infinity, ordinal numbers, what is cardinality, ordinal vs cardinal numbers, bigger than infinity, infinity is bigger than you think

Id: Szd6RBsWntg

Length: 25min 56sec (1556 seconds)

Published: Tue Nov 13 2018