How To Count Past Infinity

Hello vsauce michael, what is the largest number you can imagine? A google a google plex a billion olplex. Well, actually, the largest number is 40. Covering over 12,000 square meters of land, this 40 is made from trees strategically planted in Russia. larger than the battalion markers on Signal Hill in Calgary, the six found on Foven's badges in England, even the unrolled mile of Brady pie at number 40 is the largest number in the world in terms of area but in terms of quantity of things which is normally What we mean by a large number, 40 is probably not the largest, for example, there are 41.

Yeah, well, and then there are 42 and 43, a trillion, a trillion, you know, No matter how big a number you can imagine, you can always go higher. so there is no ultimate largest number except

infinity

. No

infinity

is not a number, but it is a type of number that you need infinite numbers to talk about and compare quantities that are endless, but some infinite quantities, some infinities are literally bigger than others, let's visit some of them. and

count

ing after them first thing first when a number refers to how many things there are is called a cardinal number, for example, 4 bananas 12 flags 20 points 20 is the cardinality of this set of points now two sets have the same cardinality when they contain the same number of things we can prove this equality by matching each member of a set one by one with each member of the other same cardinality quite simple we use the natural numbers which are 0 1 2 3 4 5 and so on as cardinal as long as we talk about how many things there are , but how many natural numbers there are, it cannot be some number in the naturals because there will always be one more that number after, instead, there is a unique name for this quantity null aleph aleph is the first letter of the Hebrew alphabet and null aleph is The first smallest infinity is how many natural numbers there are, it is also how many even numbers there are, how many odd numbers there are, it is also how many rational numbers, i.e. fractions, there are.

That may sound surprising since fractions appear more numerous on the number line, but as Cantor showed that there is a way to organize every possible rational so that the naturals can be put into one-to-one correspondence with them, they have the same cardinal point, it is aleph null, it is a large quantity greater than any finite quantity a google a googleplex a googleplex factorial a the power of a googleplex to a googleplex squared multiplied by the number of gram null aleph is greater but we can

count

beyond how well we use our old friend the supertask 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 infinite number of lines into a finite space. The number of lines here is equal to the number of natural numbers there are. Can the two be combined one by one? There is always a natural next but there is also always a next line. Both sets have cardinality off null, but what happens when I do this? How many lines are there? Aleph null plus one. There are no infinite quantities. I don't like finite quantities, there are still only null aleph lines here because I can join the natural ones one by one like before.

I started here and then continue from the beginning. Clearly, the number of lines has not changed. I can even add two more. lines three plus four plus I always end up with just null aleph stuff, I can even add another infinite null aleph of lines and still not change the amount, every even number can be paired with these and every odd number with these, there is still a line for each Naturally, another interesting way to see that these lines don't add up to the total is to show that you can do this same sequence without drawing new lines, just take every other line and move them all together to the end, it's the same thing, but Wait a second , this and this may have the same amount of stuff, but there's clearly something different about them, right?

I mean, if it's not how many things they're made of, what is it made of? Well, let's get back to just having one line later. a null size collection of aleph, what if instead of matching the naturals one by one we insist on numbering each line according to the order in which it was drawn, so we have to start here and number from left to right now what number does this line get in the kingdom? of infinity labeling things in order is quite different from counting them. You see, this line does not contribute to the total, but to label it according to the order in which it appeared, we need a set of number labels that extend beyond the natural we need ordinal numbers the first transfinite ordinal is omega the lowercase Greek letter omega this It's not a joke or a trick, it's literally just the next tag you'll need after first using the infinite collection of each county number if you got the omegath place in a race, which would mean an infinite number of people finished the race and then you did it, after omega comes omega plus one, which doesn't really look like a number, but it's like two hundred, twelve or eight hundred. comes omega plus two omega plus three ordinal numbers label things in order ordinals do not refer to how many things there are, but rather they tell us how those things are organized their type of order the type of order of a set is just the first ordinal number which is not necessary Label everything in the set in order, so that for finite numbers the cardinality and order type are the same, the order type of all naturals is omega, the order type of this sequence is omega else one and now it's omega plus two, no matter how long an arrangement it becomes.

As long as it is well ordered, as long as each part contains an initial element, everything describes a new ordinal number, this will always be very important later, you should keep in mind at this point that if you ever play a game of who can you name the largest number and you are considering saying omega plus one, you should be careful, your opponents may require that the number you name be a cardinal that refers to a quantity, these numbers refer to the same number of things, but organized in a way different, omega plus one is not bigger than omega, it just comes after omega, but aleph null is not the end, because it can be shown that there are infinities larger than aleph null that literally contain more stuff.

One of the best ways to do this is with the Cantor diagonal. I used the argument in my episode about Bonnatarsky's paradox to show that the number of real numbers is greater than the number of natural numbers, but for the purposes of this video, let's focus on something bigger than alec nol, the power set of null aleph. The power set of a set is the set of all the different subsets you can form, for example, of the set of one and two. I can do a set of nothing or one or two, four, one and two.

The power set of one, two, three is the empty set one and two and three and one and two and one and three and two and three and one two three As you can see, a power set contains many more members than the original set two raised to the number of members the original set had to be exact, then what is the power set of all natural numbers? Well, let's see, let's imagine a list of all the natural numbers. Great, now the subset of all, let's say even numbers would look like this, yes, yes, no, yes, and so on, the subset of all odd numbers.

It would look like this, here is the subset of just 3, 7, and 12. And how about every number except 5 or no number except five? Obviously, this list of subsets will be infinite, but imagine combining them all one by one with a natural. If even then there is a way to keep producing new subsets that are clearly nowhere to be found here, we will know that we have a set with more members than natural numbers, an infinity greater than aleph null, the way to do this is to start here at the first subset and just do the opposite of what we see 0 is a member of this so our new set will not contain zero then move diagonally down to membership in the second subset one is a member of it so it won't be in our new one, two is not in the third subset, so it will be in hours and so on, as you can see we are describing a subset that will, by definition, be different in at least one way from all other subsets of this Aleph. null-sized list even if we put this new subset back into diagonalization we can still make it the power set of the naturals will always resist a one-to-one correspondence with the naturals is an infinity larger than aleph null repeated applications of the power set will produce sets that cannot be put into one-to-one correspondence with the last one, so it is a great way to quickly produce larger and larger infinities.

The point is that there are more cardinals after aleph null, let's try to get to them now, remember that after omega ordinals. they divide and these numbers are no longer cardinal, they do not refer to a quantity greater than the last cardinal we reached, but maybe they can lead 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 some point down here you can always add one, we can always really talk about this never-ending process as a whole and then follow it up with something, of course We can, this is mathematics, not science, the things we assume to be true.

Mathematics is called axioms, and an axiom we come up with is no more likely to be true if it better explains or predicts what we observe. Instead, it is true because we say that its consequences simply become what we observe. We are not adapting 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 true lead us to contradictions or paradoxes, we can go back and modify them or just abandon them all. together or we can simply refuse to allow ourselves to do the things that cause the paradoxes.

That's all. The fascinating thing is that by ensuring that the axioms we accept do not lead to problems, we have turned mathematics into something that is just as the saying goes. unreasonably effective in the natural sciences, so to what extent are we inventing all this or discovering it, it is difficult to say that all we have to do to get omega is to say that there is omega and it will be good, that is what ernest zermelo did in 1908 .when he included the axiom of infinity in his list of axioms for doing things in mathematics, the axiom of infinity is simply the statement that there exists an infinite set, the set of all natural numbers, if you refuse to accept it, that's fine, that makes you a finitist. who believes that only finite things exist, but if you accept it like most mathematicians do, you can go quite a bit beyond these and through them we will eventually get to omega plus omega, except we have reached another ceiling, go to omega plus omega would be to create another infinite set and the axiom of infinity only guarantees that this one exists, we are going to have to add a new axiom every time we describe aleph null plus numbers no, the replacement axiom can help us here, this assumption states that if we take a set like let's say the set of all natural numbers and replace each element with something else like let's say bananas what you are left with is also a set that sounds simple but is incredibly useful try this take each ordinal up to omega and then instead of bananas put omega plus in front of each one now we have reached omega plus omega or omega multiplied by two using replacement we can make jumps of any size we want as long as we only use numbers that we have already achieved we can replace each ordinal up to omega with omega multiplied to reach omega multiplied by omega omega squared we are cooking now the replacement axiom allows us to construct new ordinals endlessly eventually we get to omega to omega to omega to omega to omega and we run out of standard mathematical notation no problem this is just called epsilon no and we continue from here but now think about all these ordinals all the different ways of organizing things alec null well, they are well ordered so they have a type of order 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 type of order of null things aleph, it must describe an arrangement of literally more things than the last aleph, I mean, if it didn't, it would. be somewhere here but it is not the cardinal number that describes the amount of things used to make an arrangement with type of order omega one is aleph one it is not known where the power set of the naturals falls on this line it cannot be between these cardinals because well, there are no cardinals among them, it could be equal to aleph one, that belief is called the continuum hypothesis, but it could also be larger, we simply do not know the continuum hypothesiscontinuous, by the way, is probably the biggest unanswered question in this entire topic and today in this video I will not solve it but I will go further and further towards greater and greater infinities.

Now, using the replacement axiom, we can take any ordinal we've already reached, such as omega, and jump from aleph to aleph. the output a al of omega or heck, why not use a larger ordinal like omega squared to construct alef omega squared alef omega omega omega omega omega our notation only allows me to add many countable omegas here, but replacement doesn't care No matter yes or no, I have a way of writing numbers, wherever I land, there will be a place with even larger numbers, allowing me to make even bigger and more numerous jumps than before.

It's all a massively accelerating feedback loop of ambiguity, and we can keep going. this thing of reaching ever larger infinities from below, replacement and repeated power sets that may or may not align with the aleps can keep our ascent forever, so clearly that there is nothing beyond them, not so fast, that is what we said about passing the finite to omega. Why not accept as axiom that there exists a next number so large that no amount of replacement or configuration of energy in something smaller could reach that number? called cardinal inaccessible because it cannot be reached from below now, interestingly, within the numbers we have already reached a shadow of such a number can be found aleph null this number cannot be reached from below either all numbers less than it finite and a finite number of finite numbers cannot be added multiplied exponentiated replaced with finite jumps a finite number of times or even set power a finite number of times to give you anything but another sure finite amount the power set from a millimillion to a googleplex to googleplex to googleplex is really big, but it's still finite, not even close to aleph null, the first smallest infinity, for this reason aleph null is often considered an inaccessible number, some authors do not do this, although they say that an inaccessible must also be. uncountable, which is fine, it makes sense, I mean, we've already accessed alec null, but remember that the only way to do that is to directly declare its existence axiomatically.

We will have to do the same with the inaccessible cardinals. It's really hard to understand how unfathomable the size is. of an inaccessible cardinal is, I will leave it at that, 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 larger than inaccessible ones, each of which requires a new large cardinal axiom that affirms its existence by expanding the height of our universe of numbers. Will there ever come a point where we will devise an axiom that implies the existence Of so many things that imply contradictory things, one day we will respond to the continuum hypothesis, maybe not, but there are promising directions and until then?

The surprising thing is that many of these infinities are perhaps all so big that it's not exactly clear if they actually exist or if they could be proven to exist in the physical universe. If they do, if one day physics finds a use for them, that's great. , but if No, that's cool too, that would mean that we have with this brain a tiny thing septillion times smaller than the little planet it lives on. We discover something true outside the physical realm, something that applies to the real world but is also strong enough to go beyond. beyond what even the universe itself can contain, show us or be and as always thanks for watching oh another interesting fact about transfinite ordinals is that the arithmetic with them is a little different, usually two plus one is what Same as one plus two, but omega plus. one is not the same as one plus omega one plus omega is actually just omega, think of them as order types, one thing placed before omega just consumes all the naturals and leaves us with the order type omega, one thing placed after omega requires all the natural numbers and then omega leaves us with omega plus one as the order type