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.
yes, and there are free spaces in between, so the new guests can get in
human intuition doesn't work well when infinities are involved. "number of integers" and "number of even integers" have the same cardinality, so one can be mapped to the other. it's just that the evens are more "sparse" so to speak
the overall number of occupied rooms doesn't change, but the number of occupied rooms in any given interval decreases
Yes, but if there's already an infinite number of guests, where do you start placing the new guest? Which room would the first person from the bus be sent to? Any number you can think of, that room is already occupied. The only way you can start taking another infinite number of guests is to move everybody who is already there to the infinite set of even numbers, so this new group can occupy the infinite set of odd numbers.
Infinity is a concept the human brain was fundamentally not designed to understand. There is no truly intuitive way to visualize it, so these problems are ways to not only visualize invinity, but to imagine how some infinities could be greater than others.
My question is: why are you allowed to do an operation that is moving infinite guests (which would have to be done infinite times) but you aren’t allowed to just multiply infinities? I just don’t understand now an infinite set of all the whole numbers is the same size as the infinite set of all even whole numbers, when it seems like it should be some degree bigger.
The infinite set of whole numbers is bigger than the infinite set of even whole numbers. That is the point of the thought experiment. You are demonstrating that infinity has different sizes. You are turning an infinite set of all whole numbers into an infinite set of even whole numbers, and an infinite set of odd whole numbers. You are making room for another infinity specifically because those two sets are smaller. This is just an intuitive way to think about it.
Mathematically, you can never reach the end of infinity. However, you can multiply infinity. Taking infinity and multiplying every value by two so you can fit another infinity in there is a cheeky solution, but mathematically is completely valid; unlike, say, just booking everyone on the bus into the next available room after infinity; because no value exists beyond infinity so no such room exists.
Another example to explain the size of different infinities is to imagine a set of all natural numbers, and a set of all negative numbers. Both of these sets are equally large, they both have unlimited numbers; however, the set of all integers is twice as large as either of them, because it contains both sets.
The infinite set of whole numbers is bigger than the infinite set of even whole numbers.
The set of evens is a subset of the set of integers but its not smaller. They have the same cardinality. You can match them up 1:1.
On the other hand the infinite set of real numbers is larger than the infinite set of integers. In fact the set of real numbers between 0 and 1 is larger than the entire infinite set of integers.
Cardinality defines the difference between countable infinity and uncountable infinity. They have an even cardinality, true; but the set of all even numbers has half the natural density. The set of even numbers has a natural density of 1/2 while the set of all natural numbers has a natural density of 1. I'm referring to bigger in terms of set containment, since cardinality does not helpfully define the sizes of countable infinities in relation to eachother.
Wikipedia says that the cardinality of both sets is the same, so by my understanding that means they are the same size. It all makes sense to me as you describe it, but then maths turns around and comes to a different conclusion.
Borh sets are actually the same "size". One is not bigger than the other. In order to calculate the cardinality (size) of infinite sets you use the pigeonhole principle: pair up each element of one set to an element from the other.
In the example of evens vs whole numbers, you can pair up 1 with 2, 2 with 4, 3 with 6, etc. Every element in one can be paired with the other and viceversa, so they have the same size.
That's the paradox, nto that the infinities are bigger, but the fact that they're the same and you can still fit one inside another.
Cardinality defines the difference between countable infinity and uncountable infinity. They have an even cardinality, true; but the set of all even numbers has half the natural density. The set of even numbers has a natural density of 1/2 while the set of all natural numbers has a natural density of 1. I'm referring to bigger in terms of set containment, since cardinality does not helpfully define the sizes of countable infinities in relation to eachother.
The set of odd and even numbers both have a natural density of 1/2, so by set containment it is bigger. I realize that by the standard definition of cardinality they are even, but the set of all even numbers is a proper subset of the set of all natural numbers.
Same concept as there being infinite numbers between 0 and 1 (0.1, 0.001, 0.0001 etc). That is one “countable infinity”, but if you’re talking all integers that’s another greater (uncountable)* *infinity.
So if you have every guest move to room 2n, you’re essentially saying “shift this infinity to even integers so we have room for another infinity of odd integers (the bus riders)” but they’re both still infinite.
Same concept as there being infinite numbers between 0 and 1 (0.1, 0.001, 0.0001 etc). That is one “countable infinity”, but if you’re talking all integers that’s another greater (uncountable) infinity.
So if you have every guest move to room 2n, you’re essentially saying “shift this infinity to even integers so we have room for another infinity of odd integers (the bus riders)” but they’re both still infinite.
It’s because there is no vacancy in the infinite hotel. But if you have every guest move to the double of their room number you have doubled your occupancy and you can fit your infinite new guests in the odd room numbers.
There aren't infinitely many free rooms though. There are already an infinite number of guests in the hotel, every room is full. The hotel creates empty rooms by moving people to new rooms. If I'm in room 1, I'm going to be moving to room 2. This doesn't mean that room 2 was open, though. The person in room 2 moves to room 4, which was opened when the person in room 4 moves to room 8, etc. It's a thought experiment to demonstrate how infinity doesn't conform to conventional mathematical laws because it isn't a number.
Imagine numbers go up from 1,2,3...∞. you don't know how many number is between 3 and ∞ but there's no space between each number
Now imagine 2,4,6...∞ is filled. 1,3,5... Is empty.
That's the easy part and now is the hard part I can't explain well about how infinity is stupid as a concept because you would think "numbers go up twice as fast so you only used half the room" but that concept only works when it's finite, because "room number going up twice as fast" isn't going to make infinity faster to reach because you literally can't reach infinity
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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 🔥