>>50225182Isn't it fun to think about the concept of infinity? Like, everyone knows that infinity+1=infinity, but have you ever thought about actually trying to count that precisely? It's actually easy to do! How do you count the number of objects in a set? If there's a unique correspondence between the members of that set and another, then they've got the same number of objects. That sounds stupid and redundant but it's a requirement when talking about infinite sets. For example, there are the same number of even and odd numbers because the mapping (odd)=2(even)+1 exists. The sets {0,1,2,...} and {1,2,3,...} have the same number of objects because Y=X+1 exists. Which means that (infinity)=(infinity)+1 isn't just handwavy bullshit to get you to shut up, it's an exact equality. Even more exotically, you can say that the sets {... -3, -2,-1,0,1,2,3,...} and {0,1,2,3,...} have the same number of elements even though one looks like it has 2x as many numbers as the other. That's the magic of infinity! In fact, all these sets have the same size. Call it "aleph0." The weirdest part about infinity is that there's multiple kinds of infinity. There's no unique correspondence between the discretely-spaced set {0,1,2,3,...} and the continuous set of all real numbers on the interval [0,1]. It just can't be done. So [0,1] is an infinite set that's infinitely larger than {0,1,2,3,...}. This "continuoisly-valued" infinity is called aleph1, and you can think about it like infinity-infinity, or infinity^2. Even weirder is that [0,1] has the same size as the set of all reals because the function arctan() exists. Which means that the entire real number line has the same number of numbers as [0,1], even including every number outside that interval which aren't even included in [0,1]. We can go still further beyond to construct sets that are infinity-infinity-infinity, which are infinity-times bigger than aleph1 and infinity-infinity times bigger than aleph0, but are all mutually the same size as each other just like the ones before.
Aren't numbers fun?
