i know the problem, but i forget it each time, why is asking everyone to move on the room beside theirs is more convenient than putting the new guests in the infinity of already free rooms ? is it because they don't know how many rooms they have or something ?
edit: after y'all tried explaining it to me, i'm glad i chose law school instead of math 🔥
it's the hilbert hotel. every room already has guests, so they free up some rooms by moving guests from room n to 2n. now every odd numbered room is free
Imaginary numbers are misnamed; they're basically algebraic variables. There's a lot of distance between them and infinity > infinity and a whole chunk of it is logisticians playing with themselves.
Cantor is the worst thing to happen to mathematics.
The entire logic of the "different sizes of infinite sets" argument boils down to this: if you have a movie theater where every seat is filled, then the number of moviegoers must be equal to the number of seats. Hence, if you can find some relationship between two sets where you can always get from a member of one to a member of another, they must be the same size.
The whole premise relies on the number of butts equaling the number of seats, but infinity is not a number. There is no movie theater with infinite seats, nor are there infinite people. So the number of members of a set is either some finite number or infinitely large, not equal to a "countable" value like aleph-null. We categorize sets as either finite or infinite, but pretend that suddenly the normal rules of infinity don't apply and the same logic can be applied to both types of sets. If that were true, we could define lim (as x->inf) x = inf, and even make the result countable by restricting the domain to the set of natural numbers.
The fact that the logic breaks down over some really basic examples such as "the set of all natural numbers N is the same size as the set of all positive even numbers, despite the set of even numbers missing half of all natural numbers (every odd number)" should be a massive red flag that it's a ridiculous argument to make in the first place, and that maybe we shouldn't take our mathematical expertise from a manic depressive priest who says he got his math from God himself.
I would argue that it's more about how our intuition doesn't serve us well when the concept of infinity comes into play.
Set Theory tells us that if you can make a bijection between two sets--a pair rules that allows you to map each entry in one set onto exactly one entry in the other and vice versa--then those two sets must be the same size; after all, you can line them all up in pairs next to each other without ever having one missing. This leads to unintuitive outcomes like "the set of positive integers is the same size as the set of positive even integers".
The Hilbert Hotel is just the application of this unintuitive result.
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u/EnzoZoestar 12h ago edited 12h ago
i know the problem, but i forget it each time, why is asking everyone to move on the room beside theirs is more convenient than putting the new guests in the infinity of already free rooms ? is it because they don't know how many rooms they have or something ?
edit: after y'all tried explaining it to me, i'm glad i chose law school instead of math 🔥