How To Count Past Infinity

hey Vsauce Michael here what is the biggest number you can think of a Google a googolplex a milli million all Plex well in reality the biggest number is 40 covering more than 12,000 square meters of Earth this 40 made out of strategically planted trees in Russia is larger than the battalion markers on Signal Hill in Calgary the six found on the above and badges in England even the mile of pie Brady unrolled on numberphile 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 view big 40 probably isn't the biggest for example there's 41 now and then there's 42 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 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 amounts some infinities are 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 for bananas twelve flags twenty dots twenty is the cardinality of this set of dots now two sets have 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 there'd always be one plus that number after it instead there's a unique named for this amount aleph-null Aleph is the first letter of the Hebrew alphabet and Aleph 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 may sound surprising since fractions appear more numerous on the number line but as Cantor showed there's a way to arrange every single possible rational such that the naturals can be put into a one-to-one correspondence with they have the same Cardinale point is aleph-null is a big amount bigger than any finite amount a googol a googolplex a googolplex factorial ^ a googolplex - a googolplex squared times Graham's number Aleph Nolan is bigger but we can count past it now 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 here is equal to the number of natural numbers that there are the two can be matched one two one there's always a next natural but there's also always a next line both sets have the cardinality aleph-null but what happens when I do this now how many lines are there aleph-null plus one no unending amounts aren't like finite amounts there are still only aleph-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 aleph-null things I can even add another infinite aleph-null of lines and still not change the quantity every even number can pair with these in every odd 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 what 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 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 Aleph null 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 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 by night ordinal is omega the lowercase Greek letter Omega this isn't a joke or a 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 'the place in a race that would mean that an infinite number of people finished the race and then you did after Omega comes Omega plus 1 which doesn't really look like a number but it is just like 2 or 12 or 800 then comes Omega plus 2 Omega plus 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 their 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 the same the order type of all the naturals is Omega the order type of this sequence is Omega plus 1 and now it's Omega plus 2 no matter how long an arrangement becomes as long as its 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 number and you're considering Sain Omega plus 1 you should be careful your opponents might require the number of you name to be a cardinal that refers to an amount these numbers refer to the same amount of stuff just arranged differently Omega plus 1 isn't bigger than Omega it just comes after Omega but aleph-null isn't the end why well because it can be shown that there are infinities bigger than aleph-null but literally contain more things one of the best ways to do this is with Cantor's diagonal argument in my episode on the Banach 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 Aleph null the power set of a little the power set of a set is the set of all the different subsets you can make from for example from the set of 1 & 2 I can make a set of nothing or 1 or 2 or 1 & 2 the power set of 1 2 3 is the empty set 1 & 2 & 3 & 1 & 2 & 1 & 3 & 2 & 3 & 1 2 3 as you can see a power set contains many more members than the original set to ^ 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 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 5 or no number except 5 obviously this list of subsets is going to be well infinite but imagine matching them all one to one with a 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 aleph-null the way to do this is to start up here in the first subset and just do the opposite of what we see 0 is a member of this one so our new set will not contain 0 next move diagonally down to ones 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 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 aleph-null sized list even if we put this new subset back in diagonalization can still be done the power set of the naturals will always resist a one-to-one correspondence with the naturals it's an infinity bigger than aleph-null repeated applications of power set will produce sets that can't be put into one-to-one correspondence with the last so it's a great way to quickly produce bigger and bigger infinities the point is there are more Cardinals after aleph-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 Aleph null Omega come on we've been using these numbers like there's no problem but if 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 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 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 were inventing all of this or discovering it it's hard to say all we have to do to get Omega is say let there be Omega and it will be good that's what Ernest Zermelo 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 finite list one who believes only finite things exist but if you accept it as most mathematicians do you can go pretty far past bees and through these eventually we get to Omega plus Omega except we reached another ceiling going all the way out to Omega plus Omega would be to create another infinite set and the axiom of infinity only guarantees that this one exists are we going to have to add a new axiom every time we describe out of no 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 that sounds simple but it's incredibly 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 times to 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 times Omega Omega squared we're cooking now the axiom 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 Omega and we run out of standard mathematical notation no problem this is just called epsilon not and we continue from here but now think about all of these ordinals all the different ways to arrange a love null 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 one now because by definition Omega one comes after every single order type of Elath null things it must describe an arrangement of literally more stuff than the last Aleph I mean if it didn't it would be somewhere in here but it's not the Cardinal number describing the amount of things used to make an arrangement with order type Omega 1 is Aleph 1 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 a loved one that belief is called the continuum hypothesis but it could also be larger we just don't know the 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 Aleph to Aleph all the way out to a love Omega or pick why not use a bigger ordinal like Omega square to construct Aleph Omega squared Aleph Omega Omega Omega Omega Omega Omega M in 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 rank 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 imbibing 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 ellipse 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 Omega why not accept as an axiom that there exists some next number so big no amount of replacement or power setting on anything smaller could ever get you now 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 Aleph null you can't reach this number from below either all numbers less than it are finite and a finite number of finite numbers can't be added multiplied exponentiate 'add 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 milli million to a googolplex to a googolplex to a googolplex is really big but it's still just finite not even close to Aleph null the first smallest infinity for this reason Aleph null 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 Aleph null but remember the only way we could is by straight-up declaring its existence axiomatically we will have to do the same for inaccessible 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 junk from nothing to the first infinity is like the jump from the first infinity to an inaccessible set theorists have described numbers bigger than inaccessible x' each one requiring a new large cardinal axiom asserting its existence expanding the height of our universe of 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 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 cept Elian times smaller than the tiny planet it lives on discovered something true outside of the 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 watching another interesting fact about transfinite ordinals is that arithmetic with them is a little bit different normally two plus one is the same as one plus two but Omega plus one is not the same as one plus Omega 1 plus Omega is actually just omega think about them as order types one thing placed before Omega just uses up all the naturals and leaves us with order type Omega one thing placed after Omega requires every natural number and then Omega leaving us with Omega plus 1 as the order type

Channel: Vsauce

Views: 12,701,483

Rating: 4.8961468 out of 5

Keywords: infinity, math, maths, vsauce, michael stevens, aleph, omega, transfinite, ordinals, cardinals, inaccessible, large cardinals, set theory, biggest number, largest number, biggest infinity, last number, end of numberline, types of number, number, axiom, science, theory, cantor, beyond, bigger than infinity, after infinity, counting, ordinal numbers, learn, stem, college, high school, test prep, exam, hugh woodin, woodin cardinal, woodin, vihart, the fault in our stars

Id: SrU9YDoXE88

Length: 23min 46sec (1426 seconds)

Published: Sat Apr 09 2016

Reddit Comments

[u/anooblol]

Vsauce honestly does such an amazing job of explaining these sorts of things to the average person. Sure they don't get into proofs, and it's by no means rigorous. But they do a damn good job with giving you intuition.

[u/yoloed]

I like that Vsauce's videos are starting to focus on maths.

[u/Virgilijus]

Pardon my ignorance, but hasn't it been proven that the Continuum Hypothesis is independent of ZFC? Isn't that as close to 'proving' it as we'll get?

[u/[deleted]]

This is math, not science.

12:45 I'm going to use this. A lot.

[u/[deleted]]

Would love to hear a mathematician's opinion on this.

[u/Smartless]

Can anyone tell me what to study to learn more about this kind of stuff, and what kind of background one should already have to feel comfortable studying that area?

[u/AlphaBotAlpha]

Amazing effort Vsauce always puts in their videos. Vsauce inspired me to self study set theory after watching their banach-tarski video, so glad i can understand most of what he's saying rigorously now :)

[u/[deleted]]

This video is very good, but would have been nice to touch upon the Lowenheim-Skolem theorem though.

[u/Xamimus]

Thank you for posting this. I have a set theory exam in a week with a prof that has absolutely the worst English I've heard ever. I've been pronouncing aleph-naught wrong this entire time lol.