Hey, Vsauce. Michael here. What is the biggest number you can think of? A googol? A googolplex? A milinillion?... olplex? 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 6 found on the bobbin badges in England, even the mile of pi 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 being big, 40 probably isn't the biggest. For example, there's 41. And, well, 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 can 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 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, 4 bananas. 12 flags. 20 dots. 20 is the cardinality of this set of dots. Now, 2 sets have the same cardinality when they contain the same number of things. We can demonstrate this equality by comparing 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 1+that number after it. Instead, there's a unique name 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 them. They have the same cardinality. Point is, aleph null is a big amount. Bigger than any finite amount. A googol? A googolplex? A g
