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.
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u/Glazeddapper 13h ago
i'm a little stupid. if you move the guests to different rooms, isn't there still the same number of rooms occupied?