

The Hilbert Hotel - dhotson
http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/

======
RiderOfGiraffes
There are not only different sizes of infinity, as this article is trying to
show, but also different types of infinity?

In fact, I'm coming to believe that the usual methods of teaching infinity are
wrong (not the concepts, just the presentation).

People have this intuition that there can't be anything bigger than infinity.
We reinforce that by showing that the rationals are countable, that Q^2 is
countable, in fact Q^n is countable. Even the algebraics are countable. We
create the intuition that anything infinite is countable.

Then we present 2^N and everything goes bizarre. Some people like that, but
others decide that it's all meaningless. I think this method of presentation
does the majority of people a disservice.

But not only is there the set-theoretic infinity, there is also the geometric
infinity. That actually better matches people's intuition. Separating the
ideas of the set theoretic and the geometric versions of infinity has, I'm
finding, huge benefits when trying to help people understand what's going on.

Of course, there's also cardinal versus ordinal infinities as well. That's
fun.

~~~
tokenadult
_Separating the ideas of the set theoretic and the geometric versions of
infinity has, I'm finding, huge benefits when trying to help people understand
what's going on._

Expand on that please, so I can make sure I understand what you are writing
here.

~~~
RiderOfGiraffes
It might have been you that put me onto this - I did get the idea from HN, but
haven't had time to track down the original suggestion.

This is a brief reply - I don't have time for more detail now, I am intending
to write this up, but it's about 50th on a very long "To Do" list.

People have this concept that infinity is kind of, well, "out there", as far
as you can go. You can't go any further, it's all there is. People talk about
parallel lines "meeting at infinity" and "there is no last point on the line"
sort of thing.

And that's what their intuition is telling them. There's nothing beyond
infinity.

Then when we talk about the cardinals we tend to reinforce that by showing
that the odds, evens, squares and primes all have the same cardinality. That's
a bit weird for people, but they're getting the idea that infinity is odd, but
they can cope with odd.

Then we show that N^2 and Q are both countable, can be put in one-one
correspondence with N. They're getting cool with that too. We continue to
reinforce the idea that there's infinity, and every time we do something we
get the same infinity.

But that's now what they expect. Infinity is kind of as far as you can go.
We're matching the set-theoretic cardinals to their geometrical intuition of
"out there."

No wonder they get confused, upset, and occasionally angry when we then
introduce 2^N as being "bigger."

I'm finding that talking about geometry and the concept of infinity, then
talking about sets and the concept of intinity _as a different thing, despite
the same name_ is helping people to create different models in their minds.
These different models then help them _not_ use the same intuition, and they
don't get confused in the same way.

Then, as the final modification, I don't start by showing that loads and loads
of things are all countable. I talk about "same size" as being 1-1 matching,
and discuss the idea that there could be something "bigger than N."

I do that first, and then go hunting, pointing out from the beginning that
historically people found this difficult.

Now they get excited when I show them that 2^N is uncountable. They've been
primed to want to find it, and their intuition about "infinity" isn't
challenged.

That's an incomplete summary - I hope it helps. Ask for more, or email me if
it's not clear.

~~~
TallGuyShort
I think I agree with you. I've always just thought of infinity as having no
end - and that's helped me understand all these paradoxes. For any N, 2N is
certainly bigger, even if they're both so big I could never measure either. In
my mind, infinity follows all the rules of mathematics that other numbers do
(i.e. multiplying inifinity by two ensures that it's even, IMO), it's just
that it has no end. Parallel lines don't meet at infinity - they have no
meeting point.

~~~
RiderOfGiraffes
When you write 2N do you mean "2 times N"? If you do, then your intuition is
at odds with mathematics as generally practised, because 2*aleph_0 is aleph_0.
Further, two times infinity is the same as three times infinity, and there is
no concept of infinity being even or odd. Claiming that 2 times infinity is
even can lead to contradictions and inconsistencies.

From that point of view, it would appear that your intuitions are not helping
you. More, under some models of geometry it makes a lot of sense to say that
parallel lines really do meet at infinity. It makes some theorems a lot easier
to state, and easier to prove.

So you may actually agree with a lot of what I'm saying, but your arguments to
support that claim appear, at least on the surface, to be wrong. It may be
that you have some understanding of these "paradoxes," but your other
statements suggest that your inderstanding is not that of current or classical
mathematics.

------
ynniv
Hard math is the hardest thing that I can think of. One can learn most other
things with enough effort (ie, repetition), but hard math takes immense
thought.

On the other hand, it is also rarely necessary.

~~~
npp
"Hard" is of course highly subjective, but the material in this article is
standard (usually sophomore-junior) undergrad material and is very widely
used.

~~~
btilly
Very widely used for non-useful uses of used. There aren't a lot of infinite
sets in applied math. (The closest thing that I can think of to an actual
application was <http://xkcd.com/195/> and the same trick has since been used
for visualizing other complicated large linear sets.)

~~~
npp
Most banal example is intervals in the real line (e.g. reals between 0 and 1),
which show up all the time. But comfort with infinity is needed for other uses
also, two examples being convergence and asymptotic analysis of practical
algorithms (optimization, statistical estimation, signal processing, machine
learning, geometric algorithms, ...) and showing that two classes of objects
are equivalent. The latter is useful because it can let you switch
representations of things according to which is more efficient in some
implementation, or because it shows that two seemingly different objects are
basically the same. Any class of objects that can be parameterized with a
real-valued parameter (e.g. probability distributions) forms an infinite set,
and it is often useful to be able to say things about a whole category of
problems at once, even for practical purposes.

------
stanleydrew
The actual contradiction proof that the reals have larger cardinality than the
natural numbers uses the binary representation of decimals. See Cantor's
diagonal argument: <http://en.wikipedia.org/wiki/Cantors_diagonal_argument>.

~~~
benmathes
It doesn't require binary representation at all. You can write down all the
numbers in any base and the argument still holds.

~~~
stanleydrew
Yes, clearly. I just find it interesting that in his original diagonal
argument proof he chose the decimal representation.

~~~
stanleydrew
Edit: binary not decimal.

------
davidkellis
This is a good read.

In a class I took with my thesis advisor, he explained the Hilbert Hotel
similarly, and it was as entertaining then as this is now.

------
david927
I don't believe in infinity. And most probably the reason is because it
doesn't exist.

Every time we expect something infinite in physical phenomenon, it turns out
to have some sort of Plank-like constant that compartmentalizes the effect at
different levels to avoid that result.

Which is good, because a universe where infinity existed would most likely not
be stable enough to support life. I understand that mathematics isn't simply
about physical phenomenon. I understand the beauty of i, for example. But I
think that screwing around with the bizarre aspects of a concept like infinity
is a huge waste of time.

~~~
cousin_it
_I don't believe in infinity. And most probably the reason is because it
doesn't exist._

How about the number three? Does it exist? Or the square root of 2? I can't
touch it, that's for sure.

People don't want infinity per se. We want a number system where you can do
nice things to numbers and get out other numbers... and have nice theorems
like "a continuous graph that goes from negative to positive must cross the
zero line somewhere"... and then it turns out that the number system that
fulfills our need consists of infinite sequences of digits, and finite
sequences won't suffice.

~~~
david927
Rider has set me straight, but here's my original beef:

I said, quite clearly, "I understand that mathematics isn't simply about
physical phenomenon. I understand the beauty of i, for example."

You mention 3 and the square root of 2. You plug both of those in to: (x ==
x+1) and you'll get an inequality. Not so with infinity.

Again, Rider has set me straight that I need to learn more. I can believe
that, but when I see how often in the physical world we creep up on infinity
only to have it not exist, I'm definitely skeptical about plugging in infinity
into equations like x == x + 1. It seems it doesn't make any sense at all.

~~~
cousin_it
It's true that you can't extend the number system to include something called
"infinity" that would behave like a number. Just like you can't extend the
number system to make division by zero make sense while keeping all the usual
laws of arithmetic. But this doesn't lead to any philosophical implications
like "infinity doesn't exist". In every context where it makes sense to talk
about infinities without leading to contradictions, we want to do that,
because it enriches our vocabulary and allows us to solve more problems.

Allow me to illustrate. Take the following problem stated in school geometry
terms: A square is cut into triangles of equal area, now prove that their
number is even. Interestingly, we don't yet know any elementary solution to
this problem that can be stated in school math terms. The only solution known
was discovered in the 1970s and relies on something called "p-adic numbers".
What are those p-adic numbers you ask? They are weird number-like things that
have infinitely many digits to the _left_ of the decimal point. Freaky
constructs without any counterpart in the real world, but you can still add
them, multiply them and all that. Do such things "really exist"? Don't know,
don't care. But they helped us solve a difficult problem and that's all that
matters.

~~~
david927
Cool. Thanks.

Edit: but that still seems to me to reflect infinitely, the adjective, as
opposed to infinity, the noun. Still, thanks.

------
RyanMcGreal
If you're interested in infinity, David Foster Wallace's book _Everything and
More_ is an entertaining, witty introduction.

 _Edit_ \- I'd love to know why someone downvoted this.

------
2rs1
Am I the only one that has a problem wrapping my head around the infinity
times infinity argument?

If you got infinity people on infinity busses, you would only have infinity
people, not infinity squared. The moment when you close down the first bus and
start with person 1 on bus 2, this person should already be on bus 1, or else
must bus 1 be finite.

~~~
tokenadult
The Wikipedia article

[http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gran...](http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel)

relates one method of numbering passengers on the buses to count them (which I
found in another source,

<http://faculty.cua.edu/glenn/187f09/hilbert_hotel.pdf>

which I used to teach my elementary-age class last Saturday the same trick),
which shows that countably infinite buses with countably infinite passengers
in each bus can still be accommodated by Hilbert's Hotel, even if all rooms
are occupied when the buses arrive.

------
zemaj
I've never liked these thought experiments with infinity in them. They always
have "and so on" in them. But if you're dealing with an infinite set, "and so
on" can't complete during the life of the universe (or well, ever). This in
itself seems like it solves the problem - you can never get pass the first
step. It's an event horizon, literally.

~~~
cousin_it
if you're a programmer, disbelieving in infinity is a _crippling flaw_ ,
because big-O analysis of algorithms faces the exact same kind of "event
horizon" that you speak of.

I've never liked people who don't understand math and dress it up with
philosophical gook. Math is a practical matter, not high philosophy. If you
don't know math, your bridges will fall down and your airplanes won't fly.

~~~
dagw
I wouldn't call not understanding the theoretical underpinnings of big-O
analysis a "crippling flaw" for a programmer. As long as you understand what
it means that an algorithm is O(2^n) and what limitations that entails, then
you're grand. If you can also look at an algorithm you've written and work out
what its big O complexity is then you know more than most programmers. Both of
those can trivially be done without ever touching infinity.

Equally if all you care about is working out if your bridge is going to fall
down, you don't really have any need for infinity. In fact you can spend an
entire career using math to do all kinds of really awesome things without ever
actually understanding math. I'd say that a good 95% of engineers fall into
that last category, and I'm certainly not worried about driving over bridges
because of it.

~~~
cousin_it
That's like eating meat while being morally opposed to killing animals.

~~~
dagw
How do you figure? If we want to play with food analogies I'd say more like
eating meat while not knowing how run a cattle farm.

~~~
cousin_it
While _disbelieving_ in cattle farms. That's what the discussion started from.

------
Bjoern
The infinity is a fascinating concept, but be careful it can drive you insane
and into madness [1]. At least that happend to Georg Cantor and Kurt Goedel
and some other famous minds.

[1] See "BBC: Dangerous Knowledge" e.g.
<http://www.abdn.ac.uk/modern/node/164>

~~~
RiderOfGiraffes
Much discussion about that programme here:

<http://news.ycombinator.com/item?id=797723>

<http://news.ycombinator.com/item?id=121063>

<http://news.ycombinator.com/item?id=101255>

Frankly, it's a load of crap, and shame on the BBC for having produced it.
Contemplating the mathematics of infinity didn't drive them insane, they were
already troubled. Hundreds of thousands of mathematicians happily deal with
infinity on a daily basis.

What a crock.

~~~
jgrahamc
Totally agree. That was an awful, awful program. As was the similarly awful
thing that "Infinity and Beyond" (<http://www.bbc.co.uk/programmes/b00qszch>).
The truth is that since Cantor we know that infinity is something we can work
with in an algebraic pattern and there's really no need for all these BS BBC
programs trying to make it look like something totally weird.

------
todd3834
I listen to a lot of Dr. William Lane Craig and he speaks often about
Hilbert's Hotel. Here is a pretty good video on the subject
[http://vids.myspace.com/index.cfm?fuseaction=vids.individual...](http://vids.myspace.com/index.cfm?fuseaction=vids.individual&videoid=102796188)

------
canoebuilder
_"and that you’re convinced that any particular person will be reached in a
finite number of steps."_

How can any point in an infinite set be reached in a finite number of steps?
For that to be the case, isn't the set in question necessarily a finite set
between 0 and infinity?

~~~
stanleydrew
Take the set of natural numbers. Pick any one and count to it. The set of
natural numbers has infinite cardinality, but any _particular_ member of the
set is finite and can be reached in a finite number of steps.

------
yogipatel
Did this remind anyone else of continuations and infinite streams?

------
dalore
Did you know there are different sizes of infinity?

------
fredBuddemeyer
if you want to get your head around infinity note that one of the greatest
novelists of our time, david foster wallace, wrote a great book about it and
cantor: "everything and more".

on a more sensational note, also note that wallace and cantor, both geniuses,
also both committed suicide.

~~~
gruseom
I don't think it's true that Cantor committed suicide. Wikipedia mentions no
such thing, and the bio at [http://www-groups.dcs.st-
and.ac.uk/~history/Biographies/Cant...](http://www-groups.dcs.st-
and.ac.uk/~history/Biographies/Cantor.html) says he died of a heart attack at
72.

~~~
fredBuddemeyer
you're right

[http://www.cosmolearning.com/documentaries/dangerous-
knowled...](http://www.cosmolearning.com/documentaries/dangerous-
knowledge-2008/) is wrong

