I've posed exactly these problems to children about 15 times now. Most children 10 and up I've talked to can figure out the first (one more guest) easily enough, the second (an infinite bus of new guests) with a little help, and the third (infinitely many infinite buses) if I draw a picture.
It helps a lot to be precise and use concrete examples. "Hilbert has an infinite hotel. We have to be careful with infinities, though, so I'll tell you what that means. If you think of a number zero or bigger, any number at all, Hilbert's hotel has a room with that number on it. Think of a number. ('2 million!') Yep, he's got that room. ('Six googolpillion!') Yep, he's got that room."
"A guest comes into the hotel and asks, 'Mr. Hilbert, do you have one more room?' Mr. Hilbert says, 'Sure!' He picks up his magic phone that calls everyone in the hotel at the same time and gives all the guests exactly the same message. After they do what he says, there's one room empty. What does he tell them to do?"
When they invariably come up with "Move up one room," it helps to belabor a couple of points. First, reformulate it as, "Look at your room number, add 1, and go to that room." (This helps them figure out "Multiply your room number by 2" as the answer to the second problem.) Second, dwell on who goes where, and whether it's a problem. "Where does the guest in room 0 go? ('Room 1.') Doesn't that have someone in it? ('Yes. Oh, no it doesn't, because he went to room 2!')"
No child has figured out my favorite fourth problem, but then it took mathematics until Cantor to figure it out, too.
"An uncountable group of people shows up at the hotel. Let me tell you what that means. They all have infinite name tags, all filled with As and Bs. Every possible name tag is in the group. [Give example names. Blow raspberries to do it.] The head of the group, whose name is 'AAAAAAAAAAAAAA...' [said with a blank look, trailing off] asks Mr. Hilbert if he has room in his hotel. Mr. Hilbert says 'No!' Why does he say that?"
Let them stew for a bit, and ask questions. Going on: "Mr. Hilbert says, 'OK, tell you what. If you give me a room assignment, I can always find someone you left out.' How does Mr. Hilbert do that?"
You can illustrate this with a game, using only four-letter names. Write down something like
AAAA
BBAA
BABA
AABA
"Can you find a four-letter name that's missing?" Play this a few times, and then ask, "Can you come up with an easier way that doesn't make you think of all the names in turn?" Show them how to flip the letters along the diagonal, and then extend to infinite names.
I've had 2 kids and 1 adult follow this to the end. It's always mind-blowing for them, though, no matter how far they get. I follow up with this:
"That stuff they taught you in school, that stuff a lot of people say they hate, is arithmetic. This is math."
EDIT: Come to think of it, I actually helped a friend's 11-year-old daughter decide that she didn't hate math using these problems. She's probably still bummed about being stuck doing arithmetic for now, though...
>"If an infinite set can be put into one-to-one correspondence with the natural numbers (N) it is called a countable set. Otherwise it is uncountable."[1]
This paradox hinges on the strange notion of cardinality of infinite sets. Specifically, the set of all even integers, the set of all odd integers, and the set of all integers(!) have the same cardinality, and therefore the same "size".
What's really cool is that physicists have recently been able to realize experimentally some of the room changing operations in Hilbert's hotel, using quantum systems:
It seems to me that the only thing that makes this a 'paradox' or is counterintuitive is the idea that a hotel with infinitely many rooms can be full. Is there some math concept that allows this or is it just semantics? It sounds like some sort of "quantum" effect where there's only a room there if you look for it.
What is the math concept that allows for a hotel with infinitely many rooms in the first place?
One way of (somewhat) formalizing this is to say that you have a set containing infinitely many rooms, and another set containing infinitely many tenants. Intuitively, it does not seem unreasonable that you can give every room its own tenant. Once you do this, then the hotel is "full" in the sense it is impossible to find an empty room.
Of course, we can also think about giving each tenant her own room, which also seems possible. If we can do both of these things, we say that the set of rooms is the same size (or cardinality) as the set of tenants. It is worth noting, that not all infinite sets are the same size [0]. For the purpose of Hilbert's paradox, I believe it is far to assume that we are dealing with countably infinite sets [1]. The mathematical result in Hilbert's paradox is that if you add elements to a countably infinite set, the result is still countable infitite.
can be set into a one-to-one relationship with the set
N2 = { 2, 3, 4, 5, 6,... }
by the function
f(n1:N1) -> N2 = { n1 + 1 }
(Using a notation I pulled out of my flying monkeys right at this moment.)
The first set, N1, is the state of the hotel with one guest in every room, i.e. { n | room n is occupied }. The second set is the state of the hotel after Mr. Hilbert's operation, with room 1 empty.
The meta-trick is that an infinite set can be put into a one-to-one relationship with some proper subsets of itself (that's one definition of "infinite", by the way).
I think the term full needs to be defined for an infinite hotel. Instead of defining full as occupancy equal to capacity, it might be more useful to define it as vacancy equal to 0.
Put in more concrete terms (hard to say when we are talking about infinities): there are an infinite number of integers. For each integer, there are an infinite number of real numbers (decimals) between n and n+1. For each of those doubly-infinite real numbers, there are an infinite number of complex numbers with that real component and varying imaginary components. And so on ...
In fact, there are an infinite number of infinities ...
Just to clarify a little bit: For each integer n there is also an infinite amount of rationals (fractions) between n and n+1 and the rationals is also a countable set like the integers (just as 'infinite', so to speak). Quite counter-intuitive, I think.
Also, I support her suggestion in "Infinite Trees Are Super Weird" that since we call uncountable line segments the "long line", we should call the uncountably-branching tree the "big tree".
I don't think the summary in the comment is correct, though. There are also an infinite number of rational numbers between any given integer n and n+1, and indeed an infinite number of rational numbers between any two rational numbers, yet the cardinality of the rationals is still the same as the cardinality of the integers.
It's a little confusing though, at least for me. There are infinite integers, and "doubly-infinite" reals. But there are also only "doubly-infinite" complex numbers.
IMO the easiest way to illustrate different, and infinite number of, infinities is to think of sets. Suppose you have N-infinite number of things, say triple-infinite to use the grandparent's lingo. Now the number of sets that you can create from these triple-infinite objects is greater than triple-infinite, and it's called quad-infinite. You can prove that indirectly. Assume that there is a one to one mapping of the original object to the sets. That means that some of the objects are mapped to sets that the objects are members of (let's call these red objects), others not (let's call them blue objects). Does the set of blue objects get paired with a blue or a red object ? Cannot be either, so we've reached a contradiction, ergo there are always more subsets of things than things.
Wait, but between n and n+1 there are also an infinite number of rational numbers, but the cardinality of all rational numbers is the same as that of integers (or that of all rational numbers between a fixed n and n+1).
So, while what the gp wrote is correct, it's not an explanation of why there are more real numbers than integers.
Maybe this is my ignorance of the understanding of countably infinite. But if a hotel has infinite amount of rooms, and all rooms are full. Then why would anyone ever show up to take another room.. To me it would seem that all persons are in the hotel already.
Imagine you have one person for each (positive) integer, given a unique integer ID at birth, and a hotel with countably infinite rooms, each with a unique room number.
The hotel could have someone in everyone room if every person with an even number as their ID was staying there. That is, for every room number, twice the room number is a unique even number, so there's a 1-to-1 correspondence between the number of rooms and the even integers.
You could then have someone with an odd number ID show up looking for a room, and would have to rearrange from a stay-in-half-your-ID lineup.
It's a (arguably defining) property of infinite sets that they contain a strict subset (at least one guy not in the subset) with the same "size" as the whole set.
So the evens and the integers are an example of this, with both having the same "size", even though the evens are contained in the integers.
infinite number of rooms are all occupied by a person.. then a person shows up. There is one to one ratio here... of all rooms are occupied by a person.. infinite rooms.. infinite people.. if the infinity of people are all ready in the infinity of rooms..
Then who is showing up? Everyone is already in the rooms.. So no need to worry about moving anyone.
If we change it to be numbers it still doesn't work.. If we have an infinite amount of slots.. and in each slot is a number.. and all slots are full.. how can we make room for another number..
If you say that oh well there were only even numbers in the slots.. well then what you told me isn't true.. all slots aren't full with numbers.. only even numbers..
I understand what the paradox is trying to explain about sets.. but to me it just falls apart as a metaphor.
If there were one person for every natural number, then you could put every even-numbered person in room n/2 (where n is the number for that person) and still fill up the entire hotel.
This is because even numbers are infinite, and you can find a one-to-one relation with natural numbers (just divide the even number by 2).
In that case, the infinite hotel is filled, and you still have an infinite amount of people outside it that you can accommodate (the odd-numbered ones) :)
You're making very strong assumptions about the number of people there are. How did you decide that's a countable infinity? No such assumption is made in the problem statement. If there are an uncountably infinite number of people then they wouldn't all fit in the countably infinite number of rooms, so of course there are people that can show up.
I like this paradox for it's simplicity, but there's just one aspect that cracked me up.
>Suppose the hotel is next to an ocean, and an infinite number of aircraft carriers arrive, each bearing an infinite number of coaches, each with an infinite number of passengers.
hahaha.
How would we extend that?
suppose we have an infinite number of passengers, carried by an infinite number of coaches, transported by an infinite number of aircraft carriers, shoved in by an infinite number of tsunamis, which occur on an infinite number of continents, on an infinite number of Dyson spheres...
If you're interested in this subject, can I suggest Everything and More by David Foster Wallace? (Yes, that DFW.) Sure, as some reviews describe in painful detail, some of it is mathematically sketchy. On the other hand, it's a pretty good discussion of issues around infinity, such as Zeno's (and everybody else's and their cat's) paradoxes (paradices? paradise?), and why a theory of the infinite is so darn necessary.
