Mathematician Explains Infinity in 5 Levels of Difficulty | WIRED

I'm Emily real and I'm a mathematician I've been challenged to explain the concept of infinity at five levels of increasing complexity so while the concept of infinity can seem mysterious and it's very difficult to find Infinity in the real world mathematicians have developed ways to reason very precisely about the strange properties of infinity [Music] so what do you know about Infinity I think it means that it's something that's infinite that never ends that's a great way to think about it Infinity is something that never ends we're finite the opposite of infinity refers to a process or a quantity that we could actually count all the way through at least in theory if given enough time so if you had to guess how many Skittles are in this jar I would say about like 217. 217. and if we wanted to figure out the exact number how would we find out we could like put them along divide them into pieces of five and then we could use that yeah absolutely in fact I did that before you got here and it's uh 649 Skittles here's a much harder question how many pieces of glitter do you think are in that jar maybe like 4012 I'll admit I have absolutely no idea do you think it's a finite number or an infinite number finite because I can see them all in here yeah you can see them all and in fact if we were really really really patient we could do the same thing with this as with the Skittles but here's another question you said that there's a finite amount of glitter in that jar and I agree so how many jars would we need to hold an infinite amount of glitter and then if you're amount of jars very good why do you say that because if there's unlimited pieces of glitter we need unlimited pieces of Jar so let's try and imagine infinitely many jars would they fit in this room yeah no yeah absolutely not because this room holds only a finite amount of space and in fact infinitely many jars would not even fit in something called the observable universe which is the uh portion of the universe that astronomers can see really how does that make you feel I actually feel like my brain is exploding that makes me feel like my brain is that's a wonderful question a very rich question what do you think I think maybe because as you said it was unlimited you have very good intuition so there are ways that mathematicians can build infinite collections of things and if you repeat those processes it's in fact possible to build even bigger and bigger sizes of infinity so what have you learned today about Infinity I've learned that even if it's unlimited there are many different ways of making infinity and you can never actually see it all what does infinity mean to you really anything that has no end to it yeah that's absolutely right so Infinity gets used a lot of different ways in mathematics there's a way that mathematicians think of infinity as a number just like the number 13 just like the number 10 million so wow okay so the reason that mathematicians consider Infinity to be a number is that it is a size of a set so the first example of an infinite set in mathematics is the set of all county numbers so one two three four five six seven Etc that list goes on forever that is an infinite set and to be a little bit more precise it's a countably infinite set but as a number infinity is pretty strange like what do you mean by that adding Infinities multiplying Infinities and there's a sense in which it's very similar to the arithmetic that you learned about already but it's also totally different it has some very weird properties welcome to Hilbert's Hotel unlike in order ordinary hotel has countably infinitely many rooms suppose a new guest shows up you might think that the new guest could take the room that's all the way down at the end of the hall all the way except there isn't a room like that the rooms each have a number and even though there's infinitely many rooms each room is only a finite distance away so here's how we're going to make room for the new guest I'm going to ask the guest in room one to move into room two and then we're going to ask the guest in room two to move into room three and we'll continue this all the way along it looks to me like there's space for the new guest where is it it'll be room number one room number one exactly I'm going to use this symbol for infinity but what we've just shown is that one the one new guest plus infinity is equal to the same Infinity what happens if we add a second guest would it be two plus infinity equals infinity absolutely okay so now I'm going to make this story a little more complex but there's another Hilbert's Hotel down the street and they're having plumbing issues and we need to find room for them okay they can't live together they can't live here that would be a great solution I don't know I think these people don't really get along so I need to somehow create infinitely many new rooms but I can only ask each person in the hotel to move a finite distance away right so let's take the guest who is originally in room one and move them into room two so that's creating one new space for us and now I'm going to take the guest who was originally in room two and move them into room four are you starting to see a pattern here yes you're going up one each time yeah I'm increasing by one more each time so I'm doubling the room number right so this is some of the strange arithmetic of infinity so we have two Hilbert hotels Each of which have infinitely many guests right then this is equal to infinity infinity great Hilbert's hotel is a story that mathematicians have been telling themselves for almost 100 years because it's a really visceral way to think about some of the counter-intuitive properties of the arithmetic of infinity how does Infinity come across in mathematics for you so when I'm teaching calculus and talking about Concepts like limits and derivatives those are only defined precisely with infinity teaching algebra which is meant in a different sense about number systems we deal with infinite families of numbers and their operations infinite sets are somehow very exotic they're not found so commonly in the real world but they're all over mathematics [Music] what do you know about Infinity a property of something being endless great so today we're going to focus on Infinity as a cardinality and what cardinality means is it's a size of a set what are you studying I'm studying computer science studying computer science are you taking any math courses right now yeah right now I'm taking calculus 2. calculus involves the study of functions functions are one of the most fundamental concepts in mathematics but they aren't always so clearly defined what would you say a function is I would say a function is a procedure that takes an input and does some operation and returns an output that's the computer science brain thinking right there so we want to think of a function as a procedure or a mapping between sets so a function defines a one-to-one correspondence if it defines a perfect matching between the elements of its domain set and the elements of its output set we call such functions by ejections or isomorphisms so the reason I'm so interested in this idea of a bijective function or a one-to-one correspondence that guarantees that every element of one set gets matched with an element of the other set no matter how many elements there are these bijections or these one-to-one correspondences as they help mathematicians reason about Infinity how can you compare something that is endless today we're going to think about Infinity as a cardinality which is a technical term for a number that could be a size of a and we're going to use this idea of one-to-one Correspondence to try and investigate the question of whether all infinite sets have the same size so what I've drawn here are some pictures of some of the infinite sets that appear in mathematics so the natural numbers are the prototypical example of an infinite set so the natural numbers are clearly a subset of the integers both of these are infinite sets are they the same size infinity or different size Infinities I guess the integers would there'd be more integers than natural numbers I'm going to now try and convince you that there are in fact the same size infinity and this is using this idea of a one-to-one correspondence which was applied in this context by Georg Cantor what he says is if we can match up the elements of the integers with the elements of the natural numbers so that there's nothing left over so that there's a bijective function between them then that's a proof that there's exactly as many natural numbers as there are integers start by matching zero with zero and one with one but then we want to include the negatives in the list so which natural number would we match with negative one maybe two maybe two why not because now we're starting to make progress on matching all the negatives we can match the natural number three with the integer two the natural number four with the integer minus two and do you see a pattern all of the positive integers would be odd numbers and all the negative integers would be even numbers great so now I have a much harder question so we have the same challenge again evidently there are way way way more rational numbers than there are integers does that mean there's a this is a larger infinite set than the integers what do you think by intuition I would say yes but that was the same case with the integers I would imagine there might be some bijective function for mapping natural numbers to rational numbers so I'm going to use this picture to count the rational numbers by actually counting the elements of this larger set because it'll be clearer geometrically what I've drawn in this picture is the integer lattice so Z Cross C refers to the set of all of these dots so I'll start by counting the number at the origin and you can see I'm just labeling the dots around the origin moving in a counterclockwise fashion and getting progressively further away and this process could continue but maybe by now you see the pattern that would be a little bit difficult to describe as a function oh is it for each rational number there's a pair of natural or integers that represent that rational number yeah that's exactly right and now for each pair of integers I'm going to represent it by a corresponding natural number that's what's going on with this counting and when I compose those operations what I've done is I've encoded rational numbers as natural numbers in a way that reveals that they can be no larger there are no more rational numbers than natural numbers so this slope is represented by 3 2 and 3 2 is in here as 25. exactly so that's exactly right so we were hoping to compare the size of Infinity of the rational numbers with the size of Infinity of the natural numbers so what we've done is introduced an intermediate set these pair of integer points and this proves that this size of infinity is smaller than the size of infinity since we also have an injective function the other way this size of infinity is smaller than this size of infinity so therefore they must be the same size that's wild now there's one final collection of numbers that we haven't yet discussed which are the real numbers all of the points on the number line do you think that's the same size Infinity I guess again intuition seems like it must be much larger but I don't know I haven't been on a roll Dior Cantor proved that it is impossible to count all real numbers like we've just counted the rational numbers or just counted the integers this is called the cardinality of the Continuum is uncountable what I'm going to do now is form a new real number that I guarantee is not on this list okay so here's how we do this what I'm going to do is I'm going to look at the diagonal elements so I'll highlight them this continues forever and now I'm going to form a new real number by changing all of these so if you just like added one to them then that would be something that doesn't exist in any of the other ones yes you see the idea right away so I'm going to form a new real number whose first digit is different from this one and you've already convinced yourself that this number is not on this list anywhere why is that because at every Point there's at least one change from a number in there great that's exactly right so what we've proven is that this number is missing and therefore it is impossible to define a bijection between the natural numbers and the real numbers so we've started to explore some of the counter-intuitive properties of Infinity on the one hand there are infinite sets that feel very different like the natural numbers the integers the rational numbers that nevertheless have the same size or the same infinite cardinality well there are other infinities that are larger so there's more than one size of infinity not all infinities are created equal I was wondering what the kind of practical implications are what you can do with this sort of knowledge really glad you asked me that there's a practical implication for computer science Alan Turing he came up with a mathematical model of a computer something called a Terrain machine so Turin was wondering is it possible to compute every real number an arbitrary real number to within arbitrary Precision in finite time he defined a real number to be computable if you could calculate its value maybe not exactly but as accurately as you'd like in a finite amount of time and because there are uncountably infinitely many real numbers but only countably infinitely many terrain machines what that means is that the vast majority of real numbers are uncomputable so we'll never be able to access them with a computer program [Music] you're a PhD student is that right yes I'm a second year PhD student at the University of Maryland does Infinity come up in your mathematics that you're studying one place Infinity comes up is in algebraic geometry normally we think okay well if you have two lines like this you'd keep drawing them well the intersite right here but in projective space two parallel lines will also intersect at the point at infinity infinity is like this perfect concept for what we can add to a space that allows lines to have this more uniform property what's your research in so one of my main research areas is something called category Theory that's been described as the mathematics of mathematics it's a language that can be used to prove very general theorems and an interesting aspect of being a researcher in category theory that doesn't come up as much in other areas is that we have to really pay attention to the axioms of set theory in our work when you're approving theorems have you ever used the Axiom of choice yeah it's basically this idea that you can put a choice function on any set and a choice function does what exactly yeah that's a good question so the way I think about it is if you have an infinite or an arbitrary family of sets and you know for sure that none of these sets are empty then a choice function would allow you to select an element from each set sort of all at once when you've used the Axiom of choice in proofs you know which incarnation of this you've used yeah I've used it like that I've also used it in Zorn Zama and in the well ordering principle so there are three well-known famous equivalent forms of the axing of choice the well-ordering principle is the Assumption the Axiom that any set can be well ordered but there are lots of subsets of real numbers that do not have a minimal element so that ordering is not a well ordering so here's the key question do you believe the Axiom of choice I do believe The Accidental choice you do believe the eczema choice though it leads us to some strange conclusions so if the Axiom choice is true then it's necessarily the case that there exists a well ordering of the real and what that means is that we can perform induction over real numbers like we perform induction over the natural numbers this is trans-finite induction it would work for any ordinal so there must be some uncountably infinite ordinal that represents the order type of the real numbers and this allows us to prove some crazy things imagine three-dimensional euclidean space so the space that we live in extending infinitely in all directions so it is possible to completely cover three-dimensional euclidean Space by disjoint circles so infinitesimal circles disjoint Circles of radius one so what that means is you can put a circle somewhere in space and then put a second Circle somewhere in space that can't intersect with the first one because these are you know solid circles and then another Circle and somehow cover every single point in space with no gaps that's crazy yeah it's crazy it's not the only crazy thing do you have a favorite consequence of the Axiom of choice I mean the bonoctarsky Paradox is a big one so basically it says that you can using just rigid motions I think you can take one ball a solid ball with a finite volume yeah cut it up and then rearrange the pieces so that in the end you get two balls which are the exact same size the exact same volume so you've actually taken one thing and using just pretty normal operations to it you can double it which seems pretty implausible in real life right that seems crazy to me and yet it's irrefutable consequence of this Axiom that you tell me you believe is true yeah so how many infinities are there well definitely uncountably many infinities right so there's no there's certainly no stop to this procedure but could you give a precise cardinal quality to that probably not because if I could there would be a set of all sets right so cantor's diagonal argument can be abstracted and then generalized to prove that for an arbitrary set a its power set has a strictly larger cardinality and since that's true for any set we can just iterate this process when set theory was being discovered or invented or created in the late 19th century one of the natural question to ask is can there be a universe of all sets this comes up in my research in category Theory because even though there is no set of all sets we would really like for there to be a category of so what category theorists need to do to make their work rigorous is to add additional axioms to set theory one of my favorites was introduced by an algebraic geometer Alexander growthendik this is something that we sometimes call a growth indicate Universe they're also an inaccessible Cardinal it's an infinite number that is so big that it cannot be accessed by any of the other constructions within set theory it's so big that we'll never get to it and this allows us to contemplate the collection of all sets whose cardinality is bounded by this size that will never reach so you're just making a cutoff Point you're saying we're never going to get sets bigger than this anyway so we might as well make our category only include things lower smaller than that that's right so a rigorous way to work with a category of sets is to demand that it's a category of sets whose size is bounded by this cardinality Alpha say that is then an example of a category that fits into another even still larger growth in geek Universe beta so implicitly in a lot of my research I have to add an additional assumption that there exists maybe accountably many inaccessible Cardinals examples of infinite sets abound in mathematics you know we see them every day so do those Infinities exist I think you'll get a different answer from every person every mathematician you meet it is a construct so it exists in the same way that things like that like the Poetry exists when you talk about even cardinality and it's just like well here's an infinite Hotel I had one student who was like no no no it's not that does not exist when I describe like well imagine you do this infinitely many times they're done with me because they're like I can't no one can do this infinitely many times right these interesting paradoxes that come from like the ape typing in a typewriter and eventually getting to Hamlet is an example of like well if you give something forever and any random event is going to happen it can be generative yeah sure it's definitely a really interesting really interesting thing to try to talk to students about I mean I'll grant you that Hilbert's Hotel does not exist for me infinite objects absolutely exist and I can't read the thoughts in your head but I have a high degree of confidence that we have a lot of the same ideas about Infinity it's this idea that are things that you can think of do they exist getting into philosophy of math now which is exciting I mean I think that's another common misconception about mathematics is that it's so far removed from the humanities for instance I mean it's hard to ignore some of these philosophical questions particularly when we're talking about certain certain things like infinity and I think one of the most difficult things to really be precise about and to explain to students is the Continuum hypothesis what do you say to students about the Continuum hypothesis the most fun thing to teach when you teach about Infinity when students realize that you are talking about different sizes of Infinity but then a natural thing is for them to think about like what is the next size of infinity that I can think about and sort of the Continuum hypothesis is sort of one of these really hard things to grasp so what's so fascinating about the Continuum hypothesis is you take a subset of the real line that's infinite does it necessarily have either the cardinality of the naturals or the cardinality of Continuum or is there some sort of third possibility what's very surprising is the Continuum hypothesis has been complete completely resolved in the sense that we now know for absolute certain that we will never know whether it's true or false so this is a little bit confusing the standard foundational axioms of mathematics that we take for granted are completely insufficient to prove the Continuum hypothesis one way or the other you know mathematicians among other things have been very clear about exactly what they're taking as an assumption and exactly what they're concluding from it so mathematical practices to be exactly transparent about the hypotheses you need to prove your theorem so now I think of a proof of a theorem more like constructing a function where the domain of that function is all of the hypotheses that I'm assuming and then the target of that function is maybe a particular element in some universe that is the moduli space of the statement that I'm trying to prove or something like this if the foundations were to change if set theory were replaced by something else maybe dependent type Theory do you think the theorems you've proven would still be true there's a lot of math that we sort of take for granted as this is the thing that you can do without really admitting that we are creating the foundations that are the basis for the work we do later and so yes I think that if we change the foundations we would change mathematics but I think that's also very humbling in that it's not that we're sort of discovering a universal truth is we're humans constructing meaning right it's abstract art in a sense there is something there even if you can't see all the pieces for particular things and I think that's really fascinating I was thinking about this on the drive here the way that I interact with infinity I mentioned earlier is is sometimes we in number Theory especially we say well does this type of equation have infinitely many solutions and then you know the question is is are there infinitely many are there not or like you know are there infinitely money twin primes these are sort of interesting ideas but I don't think that knowing if it's infinite or not is necessarily the most interesting thing for me what's been most interesting to me is all the math that gets developed to be able to answer that question giving current technology and who knows what mathematics will look like in 100 years 150 years ago when we barely knew infinity and look where we are today yeah Infinity inspires me to imagine a world that is so much broader than what I'll ever experience with my senses or over the span of a human life the ideas can just go on and on and on forever

2023-02-04 07:53

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