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Sizes of Infinity
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Hello there! Lovely to meet you. I've been reading a lot about infinity lately and guys it is so cool. We all kind of have a concept of infinity, like something that goes on forever and ever and ever. If you try to count to the biggest number you can think of you can always add one more. The idea itself isn't that hard to grasp, but when you try to make it rigorous like mathematics likes to do that's where things get tricky. It turns out infinity comes in different sizes. This
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german guy named Georg Cantor found that there are different kinds of infinity and some are bigger than others. To understand this we actually need a lot of background information but the background information is really cool too. We'll start with Set Theory. A set is basically a group of things. This is a set of four dogs. This is a set of four forks. The number of things or elements in a set is called its cardinality, so these sets both have a cardinality of
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four. Now when two sets have the same cardinality they're said to be the same size. The size is defined in a bit of a strange way. The normal intuitive way to think about it would be to say that both of these sets have four elements, therefore they're the same size. This definition works well for finite sets but when we start dealing with infinite sets it isn't good enough. So same size is defined as the ability for the elements of both sets to be
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paired up with each other. You'll see why this is important... right now! Let's look at a set with infinite cardinality: the set of natural numbers. It starts at one or sometimes zero depending on what text book you're reading, and increases by one forever and ever and ever. This is totally allowed in set theory by the way, we're not breaking any rules. Now let's look at another infinite set: the set of squares of the natural numbers. Every element in the second set, the set
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of squares, can be paired to a number in the first set, the set of natural numbers. So then according to our definition they're the same size. But intuitively this feels wrong. There are so many more numbers in the set of natural numbers than the set of squares. Is one set bigger than the other? But they're both infinite... infinity is infinity, right? It was at this point that Georg Cantor came along to save the day. He thought a lot about infinities and actually legend
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has it that he thought about them so much that he drove himself insane, spending his last 20 years in and out of mental institutes. But here, we'll remember him for his amazing work on numbers. He introduced the idea of an infinitely enumerable set. This just means a set whose elements can be paired off with the natural numbers. The word typically used is "counted" because effectively that's what pairing things off with the natural numbers is, but that's kind of
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confusing because you can't really count all the natural numbers. You'd die around three billion. So some prefer to call it "listing" and I rather like this. Cantor declared that infinitely enumerable sets are all the same size. The set of natural numbers is obviously enumerable, as it can be paired off with the set of natural numbers. The set of squares is enumerable, as we showed before. So then it's the same size as the set of natural numbers. But are there some infinite sets
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that can't be listed? Are there different sizes of infinity? Cantor suspected there were. In particular he wondered about the relationship between the set of natural numbers and the set of real numbers. Okay we need to take a break from set theory to talk about number theory, but don't worry, it's just as fascinating. To understand Cantor's work we first need to understand the different types of numbers and their relationship to each other. We've already
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talked about the natural numbers, whole numbers from 1 or 0 to infinity. Then there are the integers. These are the natural numbers with the addition of the negatives. They increase all the way to infinity and decrease all the way to negative infinity. Then there are the rational numbers. These are just numbers that can be represented by a ratio of two integers. You might know them as fractions. All rational numbers can also be expressed as a decimal with their
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either being a finite number of decimal places or a repeating pattern. Then there are the irrational numbers, which can't be expressed as a ratio of two numbers. Root two is a common example and here's the proof of its irrationality if you're interested. I think it's pretty clever. If you express an irrational number as a decimal the digits keep going on forever and ever with no apparent pattern. They're also sometimes called surds which is actually from the word absurd
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because the ancient Greeks really didn't like them. According to legend Pythagoras' disciple Hippasus was the one who figured out that root 2 is irrational and to celebrate his finding the ancient Greeks threw him into the Mediterranean Sea, like, to drown. Yeah they didn't like irrationals. Then there are the algebraic numbers. These are just the numbers which are solutions to algebraic equations, you know, these things. They
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have one variable, usually x, and a series of terms that sum to 0, in this case, three. Negative 3 is a solution to this equation so it's an algebraic number. All integers and rational numbers are algebraic numbers, a lot of irrationals are too. So are there numbers that aren't solutions to any equation like this? Well yes there are. In fact two of the most famous numbers, pi and Euler's number e can't be calculated by any equation. These types
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of numbers have the coolest name of all: transcendental numbers, called so because they transcend the algebraic. All of the numbers I've talked about make up the real numbers. Here's a diagram to clarify things. There are also the imaginary numbers but they're not important here. Okay then now that we know about the reals and all of the numbers they're made of we can get back to infinity. Now Cantor wanted to know whether the set of real numbers was enumerable. That was,
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could every real number be paired off with a natural number just like the set of squares. Well let's take small steps shall, we but like not in a patronizing way... Can the set of integers be paired off with the natural numbers? This isn't the usual way one would write out the integers but if we keep going with this pattern we'll eventually write them all out, so yes the set of integers is infinitely enumerable. How about the set of rational numbers? Surprisingly they
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too are enumerable. If we write them out in this rather unconventional way we can see that with the first element, the numerator and denominator add up to 2. The next two elements have numerators and denominators that add up to 3. The next three elements numerators and denominators add up to 4. A list that continues like this contains all the positive rational numbers. We can just include the negative rational numbers too if we alternate like
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we did with the integers. So then the set of rational numbers are enumerable. Cantor even proved that all the algebraic numbers are enumerable. So then what about the Reals, which encapsulate both the algebraic and the transcendental? In the same paper he proved that the real numbers are not enumerable, they are in fact a different kind of infinity, imaginatively called non-enumerable. Cantor was a huge fan of proof by contradiction and he had an
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inkling that the Reals were non- enumerable, so of course, he started the proof with the assumption that they are. Imagine that each of the real numbers is represented by an omega term, this funny looking w. So if we assume that they're enumerable we can pair them off with each of the natural numbers. Cantor showed that this list will always be incomplete. Pick any number, alpha and any larger number, beta, and mark them out on the real number line. This is just a list
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of real numbers marked out on a line. Now find the first real number on your list that is bigger than alpha and the first real number that's smaller than beta, we'll call them alpha prime and beta prime. Do this a couple more times. Now the thing is you'll always be able to find two more numbers between the last two numbers. Suppose you get stuck, where v indicates around a trillion prime marks, but a finite number. Now no matter how hard you search through the list of
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enumerated real numbers you can't find another one larger than alpha v and smaller than beta v, so then are there no more reals between them? Well what about alpha v + beta V on 2, the average? Your list is missing that number. And then what about all the numbers between alpha v and the average and beta v and the average? This could go on forever, it doesn't ever stop. There are an infinity of numbers between alpha and beta and between alpha v and beta v. This
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is called the infinity of the continuum and we can't pair it off with the natural numbers the way we could with the integers and the rationals. There are two types of infinities, the enumerable infinity of the natural numbers and the non enumerable infinity of the continuum. I really like this proof because I think it conveys the intuition really well but it is also pretty abstract, so there's another proof I want to show you too, you know, just for fun. It's called proof by
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diagonalization, or the diagonalization proof, or diagonalizing, it involves diagonals. Cantor again begins by assuming that you can number all the real numbers on a list. In fact he restricts it to just real numbers between 0 and 1. We can't really do this in order because the first number would be a point followed by an infinite amount of 0s with a 1 at the end, which would take an infinite amount of paper and ink to write, so let's just choose
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randomly. These numbers are pretty good. It's got a quarter, a third, pi on 10, e on 10. It's infinite but is it complete? Nope, and this is how Cantor proved it. He went down the list in a diagonal fashion selecting the appropriate digit, for example the first digit of the first number, the second digit of the second number, the third digit of the third number and so on, and then he used them to create a new number. Then he added 1 to every digit. Now is this new number on
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the original list? Well let's do a rigorous check. Is it the first number? No because the first digit of the first number is 1 and the first digit of the second number is 2. All right then is it the second number? Well no because the second digit of the second number is 5 and the second digit of the new number is 6. It's not any of the numbers. We've found a number that isn't on the original list, so let's add it on. Now is the list complete? Well you can just
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repeat the process and find another number that wasn't on the list. By coming up with new digits in this diagonal fashion we can always find another number that isn't on the original list, therefore our list can never be complete. There are always numbers that aren't on it. It's impossible to list all the real numbers between 0 & 1, let alone 0 and infinity. Cantor also showed that not only are infinities different, some are infinitely bigger than others. The cardinality of an infinitely
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enumerable set is smaller than the cardinality of the continuum, but like, a lot smaller. But the only difference between the algebraic numbers which are enumerable and the Reals which aren't are the transcendental numbers. So what does that mean? It means that transcendental numbers make up pretty much all possible numbers! Now I want to wrap up with a quote from the book I got most of my information from. "As humans we value neatness order and patterns. We're
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interested only in numbers that have meaning to us. From counting farm animals we have invented the natural numbers, from measurement we have invented the rational numbers, and from higher mathematics we have invented the algebraic numbers. We have plucked all these numbers from the continuum while ignoring the vast depth in which they swim like microscopic bacteria in the ocean. Virtually every number is transcendental". I don't know I think
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that's pretty cool. I must say this was probably my favorite video to make because I found the topic so fascinating, but it was also rather difficult. There were a lot of subtleties which took a while to sink in. Reading through theory is one thing but if you really want to ground your understanding nothing beats working through practical problems like the ones on brilliant org. That's what I did while I was researching this video. Brilliant is a learning website which focuses on
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math, physics and computer science. There's an awesome course on number theory which extremely conveniently had a whole section on infinity. It has a lot of concepts that were included in this video like cardinality and the different kinds of infinity, and a lot that weren't like showing infinite series through geometry, a way I'd never seen it taught before. It was actually so cool to see a visual representation like that. A subscription
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will get you access to all of these courses. Signing up is free and brilliant is offering a 20% discount to their premium program to the first 200 people who sign up using this link. Just go to brilliant.org/upandatom, link is on screen and in the description. Thanks so much for watching guys! I know this video was pretty long but I physically could not not include stuff it was all so fascinating. It was actually a requested topic by one of my patrons Robert, hi
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Robert! I hope you enjoyed watching the video as much as I enjoyed creating it. A big thank you to all my patrons for your ongoing support and until next time, bye!
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