Don't worry, you can never be put in a room with an infinite number, only an arbitrarily large one. That's the idea of the hotel; it has an infinite number of rooms, but every room has a finite number, since you can count forever without reaching infinity.
Graham’s number is big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to Graham’s number.
Thedogdrinkscoffee@lemmy.ca
on 28 Apr 02:55
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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.
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.
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.
Thedogdrinkscoffee@lemmy.ca
on 28 Apr 12:33
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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.
Thedogdrinkscoffee@lemmy.ca
on 28 Apr 13:23
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Thanks. My mistake was to view infinite as stretching without end in both directions. Today I learned. Thanks.
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.)
TREE is an extremely fast-growing function in set theory. TREE(1) equals 1, TREE(2) equals 3, and TREE(3) equals a number so large that its lower bound easily dwarfs Graham’s Number.
DarkCloud@lemmy.world
on 27 Apr 20:21
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Just put a sign on the door telling everyone who comes there to go to the inverse wing. Or go set up in the inverse wing of the hotel yourself.
Everyone in the place is moving right along the infinite axis… But there’s a whole empty wing of the hotel if you go left at the front desk.
bratorange@feddit.org
on 27 Apr 20:41
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man hes gone have to do a lot of walking again… Offtopic: The funny thing is TREE(3) is an absurdly long distance, independent of the physical unit. Even TREE(3) times the planck length is unimaginable long, as the ratio between a meter and the planck length is absolutely neglectable against such super large numbers.
It's so large that the number of digits it has is too large to represent, and the number of digits in that number is too large to represent, and so on for a number of times that is also far too large to represent.
So the only way it’s worth taking about it is in memes right? Right?
woodenghost@hexbear.net
on 27 Apr 22:16
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Well, he should consider himself lucky, since he’s still in one of the early rooms. In fact, almost as good as the first one, since the amount of rooms with numbers smaller than 2×TREE(3) as a percentage of all the rooms, is zero. Almost every other guest has it worse.
threaded - newest
Let’s say you have to move to the infinite room. How can someone check out of the hotel?
There is no infinity room, but there is an infinity pool
You can check out any time you like
The only issue is that if I have to travel to an infinite distance to the lobby, I’ll never check out.
Don't worry, you can never be put in a room with an infinite number, only an arbitrarily large one. That's the idea of the hotel; it has an infinite number of rooms, but every room has a finite number, since you can count forever without reaching infinity.
Whew, I was worried as the infinite passanger on that bus that I’d be put into an infinite room.
Just move halfway to the lobby, then again halfway to the lobby and…
…shit.
There is no infinite room though?? What?
That’s my secret, I’m always moving.
I don’t get it. Is “tree” some kind of notation?
Yes, TREE(3) is larger than a googolplex or even graham’s number.
What about our friend TREE(Graham’s Number)
Numberphile did a video on TREE(g~64~) so you gotta go bigger.
BB(TREE(3)) :)
TREE(Rayo(10^100^))
Needs more recursion
Also is your username from stormlight archive?
:)
Justice for phenadora 😭
#fuckmoash
TREE(yomomma)
Haven’t seen them in a while. Hope they’re doing well
Why do I always feel like I need a PhD to understand even the first paragraph of Wikipedia articles about math. Is that just me?
yeah where’s the Simple English Wikipedia article for Graham’s number
Go get your PhD and write one yourself!
Graham’s number is big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts to Graham’s number.
That article is not comprehensible to most people
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.
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.
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.
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.
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.
Half of infinity is still infinity, but smaller?
Can you hand me that spatula? I need to scrape my brains off the walls.
TREE is an extremely fast-growing function in set theory. TREE(1) equals 1, TREE(2) equals 3, and TREE(3) equals a number so large that its lower bound easily dwarfs Graham’s Number.
.
Yo you don’t gotta throw shade on Grahams number like that, it’s doing its best
Well, that escalated quickly.
Just put a sign on the door telling everyone who comes there to go to the inverse wing. Or go set up in the inverse wing of the hotel yourself.
Everyone in the place is moving right along the infinite axis… But there’s a whole empty wing of the hotel if you go left at the front desk.
man hes gone have to do a lot of walking again… Offtopic: The funny thing is TREE(3) is an absurdly long distance, independent of the physical unit. Even TREE(3) times the planck length is unimaginable long, as the ratio between a meter and the planck length is absolutely neglectable against such super large numbers.
It's so large that the number of digits it has is too large to represent, and the number of digits in that number is too large to represent, and so on for a number of times that is also far too large to represent.
So the only way it’s worth taking about it is in memes right? Right?
Well, he should consider himself lucky, since he’s still in one of the early rooms. In fact, almost as good as the first one, since the amount of rooms with numbers smaller than 2×TREE(3) as a percentage of all the rooms, is zero. Almost every other guest has it worse.
Math meme … it’s a math meme!
Is math related to science?
Hissssss
I don’t think so, science deals with reality and math are just floating around
/joke
legendary ball knowledge