How To Count Past Infinity

Summary

Join Vsauce's Michael as he delves into the fascinating world of numbers beyond our comprehension. In this captivating exploration, he takes us beyond finite numbers into the concept of infinity and the even larger numbers that surpass infinity itself. Learn about cardinal and ordinal numbers, how math axioms play a role in defining infinite sets, and the intriguing notion of inaccessible cardinals. This video challenges our understanding of numbers and explores whether these immense values can have a place in the physical universe.

Highlights

• Michael explores the largest number you can think of and how it compares to mathematical infinities. 🧠
• Infinity, Aleph Null, and the power of ordinals are explained in an engaging way. 🔄
• Using math axioms, we can imagine and define endless numbers, bigger than any physical realm. 🌌
• You can count past the traditional concept of infinity by understanding different sizes of infinity. 📏
• Even within infinity, ordinal arithmetic is unique; order matters even more than in finite realms. 🔄

Key Takeaways

• Infinity isn't a number but a concept used to describe unending amounts. 💡
• Cardinal numbers denote how many things there are, while ordinal numbers label them in order. 🔢
• There are multiple infinities, some bigger than others, like Aleph Null and beyond. 🚀
• Math axioms allow us to declare the existence of infinite sets, essential for progressing beyond finite numbers. 🧮
• Inaccessible cardinals are numbers so large they can't be reached through traditional mathematical operations. 🤯

Overview

In this episode of Vsauce, Micheal takes us on a wondrous journey into the realm of numbers that go beyond imagination. Starting with familiar numbers like a million, billion, and trillion, he quickly introduces us to the concept of infinity—not just as an unreachable endpoint but a starting point for exploring even larger 'transfinite' numbers.

The dive into mathematical infinities reveals complex ideas of cardinality and how infinite sets can differ in size, with Aleph Null representing the first 'smallest' infinity. The narrative highlights how mathematicians use axioms to create and explore new infinities, much larger than what we commonly consider when we think of endless numbers.

Micheal discusses the philosophical implications of these numbers and whether such grandiose constructs exist in the physical universe or are merely a creation of human abstraction. With engaging visuals and explanations, the episode stretches our imagination about numbers and infinity, leaving us both puzzled and inspired.

Chapters

• 00:00 - 01:00: Introduction Michael from Vsauce begins by asking about the biggest number one can think of, mentioning Google, Google Plex, and million oplex. However, he humorously states that the biggest number is actually 40, referring to a massive '40' made from strategically planted trees in Russia, which covers more than 12,000 square meters. This is larger than the Battalion markers on Signal Hill.
• 01:00 - 04:00: Cardinal Numbers and Aleph Null The chapter explores the concept of large numbers, beginning with an introduction to the theoretical surface area of Pi and contrasting it with typical understandings of large quantities. The discussion includes a playful nod to numbers like 40, 41, and 42 to emphasize how numbers can be conceived in different contexts, particularly in relation to cardinality and infinite sets like aleph null (ℵ₀).
• 04:00 - 06:00: Super Task and Infinity The chapter discusses the concept of infinity in mathematics. It begins by explaining that no matter how large a number you can conceive, there's always a larger one, implying there is no largest number. The text clarifies that infinity itself is not simply a number; rather, it is a concept used to describe quantities that have no end. The chapter aims to introduce readers to the idea that some infinities are bigger than others.
• 06:00 - 08:00: Ordinal Numbers and Omega The chapter titled 'Ordinal Numbers and Omega' explores the concept of ordinal and cardinal numbers. Cardinal numbers are used to indicate the quantity of a set, such as '20 dots,' where 20 represents the cardinality. The chapter sets the stage to delve into the intricacies of ordinal numbers, which are numbers that denote the position of an element in an ordered sequence. This foundational understanding allows for a deeper investigation into mathematical concepts and sets, paving the way for discussion on more complex topics such as the idea of Omega, a concept in set theory representing the first infinite ordinal number and serving as a gateway to understanding infinity within mathematics.
• 08:00 - 13:00: Power Sets and Larger Infinities The chapter titled 'Power Sets and Larger Infinities' delves into the concept of cardinality in set theory. It discusses that two sets have the same cardinality when their elements can be paired one-to-one with each other. The chapter uses natural numbers (0, 1, 2, 3, 4, 5, etc.) to illustrate the concept of cardinality. However, it raises the question of how many natural numbers exist, suggesting that the answer cannot simply be a natural number, leading to discussions about different sizes of infinity and the concept of infinite sets.
• 13:00 - 19:00: The Axiom of Infinity and Replacement The chapter discusses the concept of infinity, specifically focusing on 'Aleph Null,' the first smallest infinity, representing the quantity of natural numbers. It elaborates on how this same infinite quantity is also applicable to even numbers, odd numbers, and rational numbers (fractions), despite the differences in these sets.
• 19:00 - 23:00: Inaccessible Cardinals This chapter explores the concept of cardinality in mathematics, particularly focusing on infinities and how they compare to finite numbers. The term 'inaccessible cardinals' is used to describe specific types of infinity that cannot be reached merely through repeated power set operations starting from smaller infinities. The chapter dives into the surprising realization that fractions, despite seeming to be more numerous, can actually be matched one-to-one with natural numbers, meaning they share the same cardinality. This point illustrates how infinite sets can be counterintuitive compared to finite ones, exemplified by contrasting infinite numbers like 'Aleph-null' with finite gargantuan numbers such as Googleplex.
• 23:00 - 24:00: Conclusion In this final chapter, the concept of large numbers and infinity is explored through a mathematical lens. The discussion begins with the concept of a 'Google Plex' and moves into even larger numbers like 'Alf null'. This is followed by a thought experiment involving a 'super task'. This involves drawing an infinite number of lines, each smaller and closer to the next, fitting them within a finite space. This demonstrates how we can conceptualize and mathematically represent an infinite series of actions within a limited framework.

How To Count Past Infinity Transcription

• 00:00 - 00:30 hey Vsauce Michael here what is the biggest number you can think of a Google a Google Plex a million oplex well in reality the biggest number is 40 covering more than 12,000 square m of Earth this 40 made out of strategically planted trees in Russia is larger than the Battalion markers on Signal Hill in
• 00:30 - 01:00 Calgary the six found on the faven badges in England even the mile of Pi Brady unrolled on number file 40 is the biggest number on Earth in terms of surface area but in terms of amount of things which is normally what we mean by A number being big 40 probably isn't the biggest for example there's 41 well and then there's 42
• 01:00 - 01:30 and 43 a billion a trillion you know no matter how big of a number you can think of you could always go higher so there is no biggest last number except Infinity no Infinity is not a number instead it's a kind of number you need infinite numbers to talk about and compare amounts that are unending but some unending amount some infinities are
• 01:30 - 02:00 literally bigger than others let's visit some of them and count past them first things first when a number refers to how many things there are it is called a cardinal number for example four bananas 12 Flags 20 dots 20 is the cardinality of this set of dots now two sets have
• 02:00 - 02:30 the same cardinality when they contain the same number of things we can demonstrate this equality by pairing each member of One set one to one with each member of the other same cardinality pretty simple we use the natural numbers that is 0 1 2 3 4 5 and so on as Cardinals whenever we talk about how many things there are but how many natural numbers are there it can't be some number in the Naturals because
• 02:30 - 03:00 there'd always be one plus that number after it instead there's a unique name for this amount Alf null Alf is the first letter of the Hebrew alphabet and Alf null is the first smallest Infinity it's how many natural numbers there are it's also how many even numbers there are how many odd numbers there are it's also how many rational numbers that is fractions there are that Mak may sound
• 03:00 - 03:30 surprising since fractions appear more numerous on the number line but as caner showed there's a way to arrange every single possible rational such that the naturals can be put into a on toone correspondence with them they have the same cardinality point is Alf null is a big amount bigger than any finite amount a Google a Google Plex a Google Plex factorial to the power of Google Plex to
• 03:30 - 04:00 a Google Plex squared times grams number Alf null is bigger but we can count past it how well let's use our old friend the super task if we draw a bunch of lines and make each next line a fraction of the size and a fraction of the distance from each last line well we can fit an unending number of lines into a finite space the number of lines lines here is equal to
• 04:00 - 04:30 the number of natural numbers that there are the two can be matched one to one there's always a next natural but there's also always a next line both sets have the cardinality AL of null but what happens when I do this now how many lines are there Alf n + one no unending amounts aren't like finite amounts
• 04:30 - 05:00 there are still only Alf null lines here because I can match the naturals one to one just like before I just start here and then continue from the beginning clearly the amount of lines hasn't changed I can even add two more lines three more four more I always end up with only alol things I can even add another infinite alip of lines and still not change the quantity every even number compare with these and every odd
• 05:00 - 05:30 number with these there is still a line for every natural another cool way to see that these lines don't add to the total is to show that you can make this same sequence without drawing new lines at all just take every other line and move them all together to the end it's the same thing but hold on a second this and this may have the same number of things in them but clearly there's something different about them right I mean if
• 05:30 - 06:00 it's not how many things they're made of what is it well let's go back to having just one line after an alfn sized collection What If instead of matching the naturals one to one we insist on numbering each line according to the order it was drawn in so we have to start here and number left to right now what number does this line get in the realm of the infinite labeling things in
• 06:00 - 06:30 order is pretty different than counting them you see this line doesn't contribute to the total but in order to label it according to the order it appeared in well we need a set of labels of numbers that extends past the naturals we need ordinal numbers the first trans finite ordinal is Omega the lowercase Greek letter Omega this isn't a joke or a
• 06:30 - 07:00 trick it's literally just the next label you'll need after using up the infinite collection of every single counting number first if you got Omega Place in a race that would mean that an infinite number of people finished the race and then you did after Omega comes Omega + one which doesn't really look like a number but it is just like 2 or 12 or 800 then comes Omega plus 2
• 07:00 - 07:30 Omega + 3 ordinal numbers label things in order ordinals aren't about how many things there are instead they tell us how those things are arranged they order type the order type of a set is just the first ordinal number not needed to label everything in the set in order so for finite numbers cardinality and Order type are the same the order type of all the Naturals is Omega the order type of
• 07:30 - 08:00 this sequence is Omega + 1 and now it's Omega + 2 no matter how long an arrangement becomes as long as it's well ordered as long as every part of it contains a beginning element the whole thing describes a new ordinal number always this will be very important later on it should be noted at this point that if you are ever playing a game of who can name the biggest
• 08:00 - 08:30 number and you considering saying Omega + 1 you should be careful your opponents might require the number you name to be a cardinal that refers to an amount these numbers refer to the same amount of stuff just arranged differently Omega + 1 isn't bigger than Omega it just comes after Omega but Alf null isn't the end why well because it can be shown
• 08:30 - 09:00 that there are Infinities bigger than alol that literally contain more things one of the best ways to do this is with cantor's diagonal argument in my episode on the bonak tarski Paradox I used it to show that the number of real numbers is larger than the number of natural numbers but for the purposes of this video let's focus on another thing bigger than alfn the power set of Alf n
• 09:00 - 09:30 the power set of a set is the set of all the different subsets you can make from for example from the set of one and two I can make a set of nothing or one or two or 1 and two the power set of 1 two 3 is the empty set one and two and three and 1 and two and 1 and three and two and three and 1 2 3 as you can see a
• 09:30 - 10:00 power set contains many more members than the original set two to the power of however many members the original set had to be exact so what's the power set of all the naturals well let's see imagine a list of every natural number cool now the subset of all say even numbers would look like this yes no yes
• 10:00 - 10:30 no yes no and so on the subset of all odd numbers would look like this here's the subset of just 3 7 and 12 and how about every number except five or no number except five obviously this list of subsets is going to be well infinite but imagine matching them all one to one with a
• 10:30 - 11:00 natural if even then there's a way to keep producing new subsets that are clearly not listed anywhere here we will know that we've got a set with more members than there are natural numbers a bigger infinity than Al null the way to do this is to start up here in the first subset and just do the opposite of what we see Zero is a member of this one so our new set will not contain zero next
• 11:00 - 11:30 move diagonally down to one's membership in the second subset one is a member of it so it will not be in our new one two is not in the third subset so it will be in ours and so [Music] on as you can see we are describing a subset that will be by definition different in at least one way from every single other subset on this Alf null sized list even if we put this new subset back in
• 11:30 - 12:00 diagonalization can still be done the power set of the naturals will always resist a on toone correspondence with the naturals it's an Infinity bigger than Al of null repeated applications of power set will produce sets that can't be put into one toone correspondence with the last so it's a great way to quickly produce bigger and bigger Infinities the point is there are more
• 12:00 - 12:30 Cardinals after Alf null let's try to reach them now remember that after Omega ordinals split and these numbers are no longer Cardinals they don't refer to a greater amount than the last Cardinal we reached but maybe they can take us to one wait what are we doing Alf null Omega come on we've been using these numbers like there's no problem but if
• 12:30 - 13:00 at any point down here you can always add one always can we really talk about it this endless process as a totality and then follow it with something of course we can this is math not science the things we assume to be true in math are called axioms and an axiom we come up with isn't more likely to be true if it better explains or predicts what we observe instead it's true
• 13:00 - 13:30 because we say it is its consequences just become what we observe we are not fitting our theories to some physical Universe whose behavior and underlying laws would be the same whether we were here or not we are creating this universe ourselves if the axioms we declare to be true lead us to contradictions or paradoxes we can go back and tweak them
• 13:30 - 14:00 or just abandon them all together or we can just refuse to allow ourselves to do the things that cause the paradoxes that's it what's fascinating though is that in making sure the axioms we accept don't lead to problems we've made math into something that is as the saying goes unreasonably effective in the Natural Sciences so to what extent we're inventing all of this or discovering it it's hard to say all we have to do to
• 14:00 - 14:30 get Omega is say Let There Be Omega and it will be good that's what Ernest zero did in 1908 when he included the Axiom of infinity in his list of axioms for doing stuff in math the Axiom of infinity is simply the Declaration that one infinite set exists the set of all natural numbers if you refuse to accept it that's fine that makes you a finitist
• 14:30 - 15:00 one who believes only finite things exist but if you accept it as most mathematicians do you can go pretty far past these and through these eventually we get to Omega plus Omega except we've reached another caling going all the way out to Omega plus Omega would be to create another infinite set and the axium of infinity
• 15:00 - 15:30 only guarantees that this one exists are we going to have to add a new Axiom every time we describe Alf null more numbers no the Axiom of replacement can help us here this assumption states that if you take a set like say the set of all natural numbers and replace each element with something else like say bananas what you're left with is also a set it sounds simple but it's incredibly
• 15:30 - 16:00 useful try this take every ordinal up to Omega and then instead of bananas put Omega Plus in front of each now we've reached Omega plus Omega or Omega * 2 using replacement we can make jumps of any size we want so long as we only use numbers we've already achieved we can replace every ordinal up to Omega with Omega times it to reach Omega time Omega
• 16:00 - 16:30 Omega squar we're cooking now the axium of replacement allows us to construct new ordinals Without End eventually we get to Omega to the Omega to the Omega to the Omega to the Omega and we run out of standard mathematical notation no problem this is just called Epsilon KN and we continue from here but now think about all of these ordinals all the different ways to arrange alol
• 16:30 - 17:00 things well these are well ordered so they have an order type some ordinal that comes after all of them in this case that ordinal is called Omega 1 now because by definition Omega 1 comes after every single order type of alfl things it must describe an arrangement of literally more stuff than the last Al I mean if it didn't it would be someone in
• 17:00 - 17:30 here but it's not the Cardinal number describing the amount of things used to make an arrangement with order type Omega 1 is Alf one it's not known where the power set of the naturals falls on this line it can't be between these Cardinals because well there aren't Cardinals between them it could be equal to Al one that belief is called the Continuum hypothesis but it could also be larger we just don't know the
• 17:30 - 18:00 Continuum hypothesis by the way is probably the greatest unanswered question in this entire subject and today in this video I will not be solving it but I will be going higher and higher to bigger and bigger Infinities Now using the replacement Axiom we can take any ordinal we've already reached like say Omega and jump from Alf to ALF all the way out to ALF Omega or heck why not use a bigger ordinal like Omega squar to
• 18:00 - 18:30 construct Alf Omega squar Alf omega omega omega omega omega omega our notation only allows me to add countably many omegas here but replacement doesn't care about whether or not I have a way to write the numbers it reaches wherever I land will be a place of even bigger numbers allowing me to make even bigger and more numerous jumps than before the whole thing is a wildly accelerating feedback loop of
• 18:30 - 19:00 igging we can keep going like this reaching bigger and bigger Infinities From Below replacement and repeated power sets which may or may not line up with the alfs can keep our climb going forever so clearly there's nothing Beyond them right not so fast that's what we said about getting past the finite to omeg GA why not accept as an axiom that
• 19:00 - 19:30 there exists some next number so big no amount of replacement or power setting on anything smaller could ever get you there such a number is called an inaccessible Cardinal because you can't reach it from below now interestingly within the numbers we've already reached a shadow of such a number can be found Alf null you can't reach this number From Below either all numbers less than
• 19:30 - 20:00 it are finite and a finite number of finite numbers can't be added multiplied exponentiated replaced with finite jumps a finite number of times or even power set a finite number of times to give you anything but another finite amount sure the power set of a million million to a Google Plex to a Google Plex to a Google Plex is really big but it's still just finite not even close to ALF null the
• 20:00 - 20:30 first smallest Infinity for this reason alfl is often considered an inaccessible number some authors don't do this though saying an inaccessible must also be uncountable which okay makes sense I mean we've already accessed alfn but remember the only way we could is by straight up declaring its existence axiomatically we will have to do the same for inaccessible
• 20:30 - 21:00 Cardinals it's really hard to get across just how unfathomable the size of an inaccessible Cardinal is I'll just leave it at this the conceptual jump from nothing to the first Infinity is like the jump from the first Infinity to an inaccessible set theorists have described numbers bigger than inaccess each one requiring a new large Cardinal axium Asser its existence expanding the height of our universe of
• 21:00 - 21:30 numbers will there ever come a point where we devise an axiom implying the existence of so many things that it implies contradictory things will we someday answer the Continuum hypothesis maybe not but there are promising directions and until then the amazing fact remains that many of these infinities perhaps all of them are so
• 21:30 - 22:00 big it's not exactly clear whether they even truly exist or could be shown to in the physical Universe if they do if one day physics finds a use for them that's great but if not that's great too that would mean that we have with this brain a tiny thing a septian times smaller than the tiny planet it lives on discovered something true outside of the
• 22:00 - 22:30 Physical Realm something that applies to the real world but is also strong enough to go further past what even the universe itself can contain or show us or be and as always thanks for [Music] watching you wish never
• 22:30 - 23:00 [Music] there's only [Music] just you wish [Music] never there's only one just one out there you wish
• 23:00 - 23:30 another interesting fact about transfinite ordinals is that arithmetic with them is a little bit different normally 2 + 1 is the same as 1 + 2 but Omega + 1 is not the same as 1+ Omega 1 plus Omega is actually just Omega think about them as order types one thing placed before Omega just uses up all the
• 23:30 - 24:00 naturals and leaves us with order type Omega one thing placed after Omega requires every natural number and then Omega leaving us with Omega + one as the order type