I am one of them. I still can’t get past the Hotel paradox. To me an infinite number of guests cancels out an infinite number of rooms.
Infinite guests = infinite rooms
Infinity + n = infinity
To say the bus of unbound guests could just move into infinite rooms seems to give a property of rooms without limit that is not shared with the original infinite guests.
The original premize states the hotel is full. Because the only thing that matches infinite rooms are infinite guests.
Apparently I am very stupid. My sister was right all along.
Funnily enough that’s an example of two infinities that mathematically have the same cardinality (which is very often conflated with size since for general mathematical purposes that’s what it is) since you can map a bijection (i.e. every number in the first set has one and only one mapping in the second and vice versa) between the two (and it’s as simple as f(n)=2n).
And intuitively that makes about as much (or rather, little) sense as the infinite hotel.
An example of infinities with different cardinalities would be rational numbers vs natural numbers.
I didn’t actually know this (all my math knowledge comes from what my cs degree forced me to ingest) but google says yes (since natural numbers, being a countably infinite set, are apparently an example of the smallest possible cardinality of infinite sets, so any infinite subset of natural numbers is always the same cardinality.)
Infinite hotel has infinity guests. You have all the guests move down 10 rooms. Rooms 1-10 are now free. Zero to Infinity and 11 to infinity are equally infinity, since numbers extend into infinity.
In the same manner if you have one set of infinite guests occupy all the even numbered rooms, you will still have an infinite number of rooms open, because the set of all odd (and even) numbers extends infinitely. You could have the first set of infinite guests take each hundredth room (100, 200, 300, etc), then the next set take 99, 199, 299, etc. in that way you could fit 100 sets of infinite guests.
It just illustrates that infinity is not an easily intuitable concept.
What doesn’t make sense to me is infinite rooms and infinite guests and is full. You ask everyone to move down 10 rooms, why is 1-10 now free? You had infinite guests too, wouldn’t more filled rooms appear?
Or Is infinite only infinite (undefined) on the upper end, but defined on the lower? E.g. 1.
You can define the start of an infinite series, just not the end. (Except as ∞ or -∞). You could also have an infinite set that extends both ways.
0 to ∞ contains an infinite amount of numbers. But so does 11 to ∞.
More filled rooms do not “appear”, the rooms just go on without end. These is no “last” guest who moves into some previously unoccupied room. It’s just… endless. Infinite.
It really only makes sense in abstract. Our minds aren’t built to deal with infinity.
There are different “kinds” of infinity. For example, there is an infinite amount of natural numbers, and there is an infinite amount of real numbers. Still, natural numbers only make up a tiny part of real numbers, so while both are infinite, the set of real numbers is bigger. Hilbert’s Hotel is an analogy meant to convey how to deal with these different notions of infinity.
Not really. The guests move to a room with double the number, freeing up an infinite number of rooms.
So the change is from natural numbers to even numbers, freeing up odd numbers. Those infinities are the same, but you can still do this because infinities are weird.
That article is not comprehensible to most people
https://www.youtube.com/watch?v=3P6DWAwwViU
Here is a Numberphile video that describes how large a number we are talking.
Thanks, I get it now!
I am one of them. I still can’t get past the Hotel paradox. To me an infinite number of guests cancels out an infinite number of rooms.
Infinite guests = infinite rooms Infinity + n = infinity To say the bus of unbound guests could just move into infinite rooms seems to give a property of rooms without limit that is not shared with the original infinite guests.
The original premize states the hotel is full. Because the only thing that matches infinite rooms are infinite guests.
Apparently I am very stupid. My sister was right all along.
So, some infinities are bigger than others.
How many numbers are there? An infinite number.
How many even numbers are there? An infinite amount, but half the size of the first infinity.
This is how there are empty rooms in the infinitely large hotel with infinite guests.
Half of infinity is still infinity, but smaller?
Can you hand me that spatula? I need to scrape my brains off the walls.
Funnily enough that’s an example of two infinities that mathematically have the same cardinality (which is very often conflated with size since for general mathematical purposes that’s what it is) since you can map a bijection (i.e. every number in the first set has one and only one mapping in the second and vice versa) between the two (and it’s as simple as f(n)=2n).
And intuitively that makes about as much (or rather, little) sense as the infinite hotel.
An example of infinities with different cardinalities would be rational numbers vs natural numbers.
OK, thanks for the extra info. When I see that function it kinda makes sense but stops if I think about it too much.
Would even numbers and prime numbers have different cardinality?
I didn’t actually know this (all my math knowledge comes from what my cs degree forced me to ingest) but google says yes (since natural numbers, being a countably infinite set, are apparently an example of the smallest possible cardinality of infinite sets, so any infinite subset of natural numbers is always the same cardinality.)
The answets to this stackexchange question go into detail but that’s beyond what I understand without a lot of effort tbh.
OK, that makes some degree of sense in the abstract. I’ll try to hold on to it.
Thanks for checking and getting back to me. It helps grok the cardinality a bit more.
Infinite hotel has infinity guests. You have all the guests move down 10 rooms. Rooms 1-10 are now free. Zero to Infinity and 11 to infinity are equally infinity, since numbers extend into infinity.
In the same manner if you have one set of infinite guests occupy all the even numbered rooms, you will still have an infinite number of rooms open, because the set of all odd (and even) numbers extends infinitely. You could have the first set of infinite guests take each hundredth room (100, 200, 300, etc), then the next set take 99, 199, 299, etc. in that way you could fit 100 sets of infinite guests.
It just illustrates that infinity is not an easily intuitable concept.
What doesn’t make sense to me is infinite rooms and infinite guests and is full. You ask everyone to move down 10 rooms, why is 1-10 now free? You had infinite guests too, wouldn’t more filled rooms appear?
Or Is infinite only infinite (undefined) on the upper end, but defined on the lower? E.g. 1.
You can define the start of an infinite series, just not the end. (Except as ∞ or -∞). You could also have an infinite set that extends both ways.
0 to ∞ contains an infinite amount of numbers. But so does 11 to ∞.
More filled rooms do not “appear”, the rooms just go on without end. These is no “last” guest who moves into some previously unoccupied room. It’s just… endless. Infinite.
It really only makes sense in abstract. Our minds aren’t built to deal with infinity.
Thanks. My mistake was to view infinite as stretching without end in both directions. Today I learned. Thanks.
There are different “kinds” of infinity. For example, there is an infinite amount of natural numbers, and there is an infinite amount of real numbers. Still, natural numbers only make up a tiny part of real numbers, so while both are infinite, the set of real numbers is bigger. Hilbert’s Hotel is an analogy meant to convey how to deal with these different notions of infinity.
Not really. The guests move to a room with double the number, freeing up an infinite number of rooms.
So the change is from natural numbers to even numbers, freeing up odd numbers. Those infinities are the same, but you can still do this because infinities are weird.