
Hilbert's paradox of the Grand Hotel - tokenadult
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
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mnemonicsloth
A lot of mathematicians I've known feel a kind of awe for Hilbert that they
don't feel even for history-makers like Euler or Gauss. I think little stories
like this one are part of the reason why. Hilbert always makes it look easy,
even when his results are weirdly counter-intuitive.

Technically, though, there aren't any paradoxes here. It's for the purpose of
avoiding paradox that we're willing to talk about infinities in the first
place.

~~~
slackenerny
_Technically, though, there aren't any paradoxes here._

The real reason why it _is_ a paradox doesn't have much to do with notion of
infinity per se, but with problem of _infinite decision_ aka _Supertask_.

Which reminds me of an article on physics of supertasks that among other
things somewhat outlines how classical theory is more paradoxical before than
after quantization <http://th-www.if.uj.edu.pl/acta/vol36/pdf/v36p2887.pdf>
(43pp but easy reading).

~~~
Luc
I really wanted to like this PDF, but the grammar is at times so bad it is
often quite hard to parse.

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mojuba
When you ask client N to move to N+1, what happens is this client becomes
temporarily "homeless", or not occupying any room for the duration of this
procedure: packing, moving out, asking the next client to do the same and
_waiting_ until the next client packs, moves out, etc. So once you want to
make room for your new client and start the procedure, one of the existing
clients at any given time will be in the lobby, not occupying any room.

Accommodating infitine number of newly arrived guests thus means having
infinite number of poor "homeless" guys in the lobby.

So? So... Grand Hotel can accomodate exactly infinite number of guests, but
not more than that. If it's full, it's full.

Let alone the fact that infinity doesn't exist because we can't observe it.

~~~
mnemonicsloth
_infinity doesn't exist because we can't observe it_

Presumably, you are accessing this website using a some kind of personal
computer and some communications infrastructure. The engineers who designed
these systems all made frequent use of the notion of infinity, particularly in
their use of limits and calculus, when these systems were designed.

I am sure that those engineers would _love_ to hear about the equivalent (but
necessarily much more complex), infinity-free versions of calculus and limit
processes that you have obviously developed. Since infinity is the only
simplifying assumption the average engineer is ever forced to make, I'm sure
that as a group, they will be very eager to take your advice, even if it means
they have to do twice as much work.

~~~
amalcon
Infinity doesn't "exist" because the notion of "existence" doesn't apply to
mathematical constructs. We don't find them in nature; we use them to describe
things we find in nature. The number two doesn't "exist", though there are
plenty of things it models well.

There are, of course, things that are described by infinity: the energy
required to accelerate a mass to the speed of light, the number of points at
which a magnetic field can travel through a surface, etc.

~~~
mojuba
The notion of existence is applied to mathematical constructs so long as these
constructs are abstractions/generalizations of things that exist in the
physical world.

The number two is a generalization of quantity. We know there may be two
apples or two birds on the wire, and at one point we are taught quantity can
be applied to virtually antyhing.

Now, what's infinity? Another abstraction of quantity, which has many
definitions. However, every definition you take involves infinite time, e.g.
you add 1 to a number _ad infinitum_ or you accelerate a particle to the speed
of light, and because reaching precise speed of light is not possible, you
will be doing it infinitely.

So even " _we use them to describe things we find in nature_ " - even that is
not true. You can't find infinity in nature, neither anything that can be
described with infinities.

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dxjones
What is missing from this discussion of infinity is the notion that there are
multiple infinities that are different from each other.

The integers (like the hotel rooms) are countably infinite. There are only a
FINITE number of integers (or hotel rooms) between any two integers (rooms),
like between rooms 1 and 100.

The set of real numbers is a larger set than the integers. Both are infinitely
large, but the real numbers are a larger infinity. Between any two real
numbers, (even 0 and 1), there is still an INFINITE number of real numbers in
between.

The notion of a hierarchy of infinities initially seems paradoxical, but once
you know how to distinguish countably infinite from uncountably infinite, you
have the key concept.

~~~
drbaskin
The reals and the integers do have different cardinalities, but not for the
reason you imply. In fact, the rational numbers have the same cardinality (the
same infinity) as the integers, but they have the same property that you use
to characterize the reals, i.e., that between any two distinct rationals there
is an infinite number of rationals.

One way to see that there are as many integers as there are rational numbers
is just to find a way to count the rational numbers. There are a number of
ways to do this. There is a very nice and explicit way to do enumerate the
positive rationals that involves the prime factorization of the integers.
Suppose first that p is a prime number. We identify p^k with the integer
p^{2k}. For the rational number 1/p^k (p is still prime), we label it by the
integer p^{2k-1}. For a rational number p/q in lowest terms, we take the prime
factorization of p and the prime factorization of q, apply the above
identification to each factor p_i ^k and then take the product. This gives a
very explicit bijection between the positive rationals and the positive
integers.with

My favorite way, though, is just to make a grid of pairs of integers
(numerator, denominator), and then count them by spiraling outward on the
grid. This yields duplicates, but that's ok.

~~~
mnemonicsloth
_My favorite way, though, is just to make a grid of pairs of integers
(numerator, denominator), and then count them by spiraling outward on the
grid. This yields duplicates, but that's ok._

Or just enumerate the naturals in base 11. Denote 9+1 by "/". Interpret "/" as
a negative sign when it's the first or last base-10 digit, otherwise as
division. Pick an associativity.

~~~
slackenerny
Oh, that's really nifty.

I first got embarassed by not knowing of that direct way here:
[http://gowers.wordpress.com/2008/07/30/recognising-
countable...](http://gowers.wordpress.com/2008/07/30/recognising-countable-
sets/)

------
jpwagner
What is the point of the cigar section of the article?

~~~
tokenadult
_What is the point of the cigar section of the article?_

To discuss a necessary step in mathematical induction, namely the basis.

<http://en.wikipedia.org/wiki/Mathematical_induction>

