
Does infinity exist? - ColinWright
http://plus.maths.org/content/does-infinity-exist
======
drblast
My favorite thought experiment that I think is relevant here is to try to pick
a truly random number.

If you think about it, this is impossible, because the act of picking a number
limits you to a finite set of numbers that you have considered.

Put another way, you can't pick numbers from an infinite sample space because
you can't create an infinite sample space.

This has all sorts of odd implications, like the two envelope paradox:
<http://en.wikipedia.org/wiki/Two_envelopes_problem>

Enjoy!

~~~
davmre
_Put another way, you can't pick numbers from an infinite sample space because
you can't create an infinite sample space._

That's not true; there's nothing wrong with infinite sample spaces. It's just
that there's just no such thing as a uniform distribution over an infinite
space.

For example, suppose you flip a fair coin repeatedly until the first time it
comes up Heads. The number of flips you end up making could be any positive
integer, so this procedure samples from an infinite space. But not all
integers are equally likely. (in fact, each integer is half as likely as the
one before, following a geometric distribution).

~~~
alok-g
How about generating an infinite sequence of bits to generate any positive
integer (in binary) with uniform probability? Sure it will take infinite time,
but so may the method in your note. As another comment here points out, it
would take infinite time to write down a purely random number anyways.

~~~
davmre
The method I described gives nonzero probability to every positive integer.
The method you describe a) will never terminate and thus b) gives zero
probability to every positive integer (since there's no stopping condition,
even if you ever get to a particular integer you're always going to generate
another digit and move on past it).

The essential issue is that probabilities have to sum to 1, but you can't just
give the same probability to an infinite number of things because there's no
number for which p*infinity = 1. So the only way to get an infinite sum to
equal 1 is if the probabilities are unequal, and in particular the
probabilities have to go to zero in the limit (e.g. 1/2 + 1/4 + 1/8 + 1/16 +
... = 1).

------
Swizec
My favourite example of infinity is this:

"You own a hotel with an infinite number of rooms, all of which are currently
occupied. A group of four guests comes and wants to rent a room. Do you have a
room for them?"

The answer is that yes, you do in fact have room for infinitely more guests.

~~~
ajanuary
I'm gunna have to be one of those people that says "really? O_o" when
presumably everyone else clearly grasps the concept.

I get that if you had an infinite number of rooms, and an infinite number of
guests staying, you still have room for an infinite number more guests.

But in your example, surely "occupied" is, more than just an indicator of
another guest, a state of the room. You have an infinite number of occupied
rooms, but zero unoccupied rooms, so no space for more guests.

~~~
recursive
The classic approach is to move all the existing guests from room n to n+1 and
put the new party in room 1.

~~~
damoncali
Aren't there already guests in room n+1?

~~~
leviself
Yes. The statement as given doesn't work out. If you say "an infinite number
of guests and an infinite number of rooms" it works. But when every room is
endowed with the property of being occupied there is no room for more guests.

~~~
ajanuary
It brings up an interesting point, though. If I have an infinite number of
rooms and an infinite number of guests, then one might intuit that every room
is occupied.

There is a mapping of every guest to a room: guest n is in room n. Yet somehow
there exists a room that has no guest, despite being able to model both rooms
and guests with the same infinite set.

~~~
nhaehnle
There is no empty room initially. However, by having all guests move
simultaneously, you can make one room free without having any guest leave the
hotel.

Incidentally, this is one way to define infinite sets. A set is infinite if
and only if there exists a proper subset that has the same size (cardinality).

------
tikhonj
The idea of different infinities is very important to programmers. In
particular, they are the underlying reasons for undecidable problems.

You can write any valid computer program as a string of finite length from a
finite alphabet. This means the set of programs is countable. (This should not
be surprising--everything is ones and zeroes, after all, so you always end up
mapping your program to a really large natural number to use it.)

EDIT: To clarify a bit--the set of computer programs is _infinite_ and
countable. I sometimes use "countable" to mean "countably infinite" which is a
very bad habit. My only justification is that the fact that a finite set is
countable is trivial and therefore boring.

The number of functions between two countably infinite sets is _not_
countable. This is also fairly easy to see: the set X → Y can be easily mapped
to a subset of P(X × Y) where P deontes a powerset. All this says is that you
can describe any function from X to Y as a set of ordered pairs (X, Y). For
two countable sets X and Y, X × Y is also countable; if either X or Y is
infinite, X × Y is also infinite. This means that P(X × Y) is _not_ countable,
so X → Y is not countable either.

EDIT: My justification for why X → Y is uncountably infinite if either X or Y
is countably infinite (and both are countable, of course) is not correct. Just
mapping something to a subset of P(X × Y) doesn't actually tell us anything.
However, the actual fact _is_ correct even if I don't show it, so I'm going to
figure out (or, honestly, probably look up) why this is the case and amend my
post further :P.

So the number of programs we can write _is_ countable; the number of functions
we can write over interesting domains (like natural numbers or integers) _is
not_ countable. There has to be an infinite number of functions we cannot
write programs for!

So we have managed to show, in a fairly simple way, that there have to exist
undecidable problems. The real trick is that we can see this _without_ having
to construct a problem like that or even reason much about programs in
general; all we have to know is that programs are strings and have finite
lengths.

The other neat bit is that this reasoning is very easy to adapt to proofs
rather than programs. After all, mathematical proofs are also finite-length
strings! And, for example, propositions about the natural numbers are
essentially functions ℕ → {true, false}. So you can't prove all of them.

So now we also see a deep relationship between proofs and programs, without
actually doing much thinking about proofs _or_ programs.

We can also see that I'm no mathematician and either made some blatant errors
or was not very rigorous throughout :P.

~~~
Dove
I'd never thought about that before. I think you could use that argument to
prove that strong AI is impossible.

\---------------

Consider an AI to be a chat program which maps strings to strings -- all the
strings of its input over time to all the strings of its output. There are, by
the argument above, an uncountable number of such mapping programs, only a
countable number of which can actually be coded.

So, enumerate the ones that can be coded. This shouldn't even be difficult--
let's just do it alphabetically.

Now, suppose I have a strong AI called Sal. My first input to Sal is to
describe the program enumeration strategy, and instruct her to respond to any
string which contains a number by looking up the program on the table
corresponding to that number, inputting all the input she has received so far,
and then giving a response different than what it would give.

A strong AI can certainly do this.

Sal is not programmable.

~~~
Dove
In fact, that argument can be stated a lot more simply.

Take any programmable AI. Show it its own source code. Instruct it to run the
code with all the inputs it has received so far, and then return something
different.

Hence, a strong AI cannot have source code that it is capable of running.

Huh. I'd always heard that a true mind couldn't comprehend itself. I guess now
I know why.

~~~
gjm11
This argument also proves that people can't really think, because I can set
you the following challenge: "Work out what number you'd name in response to
this question, and then tell me the number one bigger than that."

~~~
lotharbot
This doesn't prove that a human "can't think". It proves that a human can't
provide a logically consistent answer to a question that doesn't have a
logically consistent answer.

This isn't quite analogous to the situation in the parent. There exists a
logically consistent answer to the question "tell me something other than what
[algorithm] would yield from [inputs]" -- it just can't be produced by that
algorithm.

~~~
gjm11
It's exactly analogous. There is a logically consistent answer to the question
"tell me something other than what Joe McBlow would do in situation X"; it's
just that Joe McBlow can't give it to you.

------
mistercow
I find this question very perplexing. I can't tell if it arises from a
misunderstanding of the concept of "existence" or just a misunderstanding of
how important infinities are in a massive number of real world applications. I
mean, two doesn't really "exist"; but we sure as hell need it if we want to
get any work done.

~~~
nhaehnle
I agree with you there. The vast majority of "philosophical" questions of type
"Does X exist?" are really quite trivial once you clearly define what you mean
by "exist".

As in your example, the number two doesn't exist in the physical universe.
There is no physical object that we can point to and say "This is the number
two". It has some physical _representations_ (such as some sequences of
electronically encoded bytes in this very comment), sure, but that's not quite
the same thing. On the other hand, the number two obviously exists in a
mathematical sense. So it really boils down to being precise about what you
mean by "existence".

It's almost the same with infinities. I say almost, because there is some
uncertainty when it comes to physics, e.g. whether the physical universe has
discrete or continuous coordinates (and the article mentions singularities,
which is a similar problem).

------
jberryman
If this interests you, I can recommend David foster Wallace's "Everything and
more"

------
logotype
BBC Horizon - How big is the universe?
<http://www.youtube.com/watch?v=Dne6rnITayI>

Interesting.

