
Hilbert's paradox of the Grand Hotel - vinnyglennon
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
======
lancefisher
And here's a great book that explains it to children:
[http://www.amazon.com/The-Cat-Numberland-Ivar-
Ekeland/dp/081...](http://www.amazon.com/The-Cat-Numberland-Ivar-
Ekeland/dp/081262744X)

~~~
ntoronto
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...

------
mfoy_
>"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".

\---

[1][http://www.math.ups.edu/~bryans/Current/Journal_Spring_1999/...](http://www.math.ups.edu/~bryans/Current/Journal_Spring_1999/mlaycock_300_s99.html)

------
semi-extrinsic
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:

[https://physics.aps.org/synopsis-
for/10.1103/PhysRevLett.115...](https://physics.aps.org/synopsis-
for/10.1103/PhysRevLett.115.160505)

------
unclenoriega
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.

~~~
gizmo686
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.

[0]
[https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)

[1] That is to say, sets which are as large as the set of natural numbers.

------
URSpider94
A related derivation of the "uncountably infinite" is Cantor diagonalization:
[https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument](https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument)

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 ...

It's turtles all the way down.

~~~
lukasb
That's the clearest, most concise explanation I've seen for how different
infinite sets can have different cardinalities. Thanks!

~~~
mikeash
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.

Diagonalization is brilliant, though!

------
rootedbox
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.

~~~
NotAPerson
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.

~~~
rootedbox
But the paradox says...

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.

~~~
epidemian
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)
:)

------
GhotiFish
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...

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mcguire
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.

------
lojack
Reminds me of the proof showing that the set of all fractions are countably
infinite.

------
tunafishman
Situations like these are where math stops being an enlightening view on
relationships between quantities and starts being a pedantic chore.

