
Are The Reals Really Uncountable? - profquail
http://rjlipton.wordpress.com/2010/01/20/are-the-reals-really-uncountable/
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leelin
The first time I saw the proof I was not satisfied because I wondered why you
couldn't use the same exact argument to show the integers are uncountable (and
back in high school I didn't grasp the quick answer, 'because only reals have
infinite digits, any particular integer only has finite digits').

I remember Mark Krusemeyer at MC took a more satisfying approach. He first
showed the power set of integers was uncountable, then showed the one-to-one /
onto mapping to the reals (and then some bit about power sets of power sets as
a way of finding ever larger and more infinite sets).

For whatever reason, thinking of the size of power sets seem a bit less
abstract than the size of reals, maybe because it feels more discrete and
computer science-ish?

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RiderOfGiraffes
The short answer is yes, and I, like the author, don't really understand why
so many people spend so much time trying to show that Cantor's proofs are
wrong.

Cantor's second proof generalises to show that given any set, X, the
collection of functions

    
    
      { f: X -> {0,1} }
    

is of size strictly larger than X.

    
    
      |{f: X -> {0,1}}| > |X|
    

The proof that the reals are uncountable is just this, with the set X as the
set of natural numbers.

~~~
yannis
One side of my brain accepts and understands the Maths but another corner of
it can never understand the paradox as to how one infinity is greater than the
other; since the reals are uncountable that is, there are strictly more real
numbers than natural numbers, and both sets are infinite.

~~~
RiderOfGiraffes
Interesting - thanks. I wonder if the problem is that people try to think of
infinity as a number that's just really, really big - bigger than all the
numbers. If you think of infinity as being the biggest thing there is, then
you've got a problem.

You (and others) are not alone. Infinity was a major problem for a very, very
long time, with many contradictions and paradoxes arising because of
insufficiently careful reasoning.

Here's another explanation:

[http://www.solipsys.co.uk/new/CantorVisitsHilbertsHotel.html...](http://www.solipsys.co.uk/new/CantorVisitsHilbertsHotel.html?HN)

and here's something some people think of as a paradox:

<http://www.solipsys.co.uk/new/BallsInBarrels.html?HN>

~~~
lmkg
I made an interesting observation recently, and I think it's not reinforced
enough in the teaching of maths: "infinite" and "infinity" have less to do
with each other than you would think. "Infinity" is a particular point in a
set. "Infinite" describes the size of a set. There's only one "infinity"[1]
but there's (infinitely!) many types of "infinite." By application of the
Sapir-Whorf hypothesis, this would be easier for people to understand if we
used more separate terms for the two concepts.

[1] Well, you can have +/-infinity... or you can make it the same infinity at
both ends. This is really the only way to go when you deal with the complex
plane instead of the real line. But regardless, the point is in topology,
infinity can be treated as singular and definite.

~~~
RiderOfGiraffes
You have an interesting and perhaps useful conceptual point, but your language
is at odds with current usage.

The concepts of infinite sizes of sets and point "at infinity" are different,
and should be separated.

The thing is, there are multiple infinities even in the sense you're thinking
about. For example, you can have one at the end of every line in the complex
plane. That gives you the projective plane. Again, in topology.

The one you're thinking of is the one-point compactification of the plane, but
it's not the only one.

~~~
lmkg
Actually, having an infinity at the end of every line on the complex plane
just gives you a simple disc. To get the projective plane, you have to
associate antipodal points, like associating +/-infinity on the real line to a
single infinity, but for every line that runs through the origin separately.
But in topology, whether these points are 'at infinity' or not is arbitrary,
because the complex plane is indistinguishable from the unit disc, until you
start adding some more structure to it like a differential structure.

And trust me, you do not want to be doing any sort of calculus on the complex
plane with anything other than a single infinity. Conformal mappings really
don't work well without that simplification. And with that simplification,
everything is magical and you can integrate rainbows to get unicorns and candy
canes.

Nonetheless, your point is valid, it's possible to have more than a single
point at infinity. However, regardless, it's more conventional to think of
infinity as a particular point with some unusual properties, and in these
cases with multiple infinities they're all basically the same. You don't have
classes of infinities the same way you have classes of infiniteness.

~~~
RiderOfGiraffes

      > Actually, having an infinity at the end of
      > every line on the complex plane just gives
      > you a simple disc.
    

Er, judging from the rest of your comment you must simply have misunderstood
me. For every line through 0 in the complex plane, take a point and define it
to be at both ends of the line. That is doing as you say, associating
antipodal points.

And I do do calculus (well, equivalent procedures) on such objects. It's clear
from your comments that you know about these things - trust me that I do as
well.

So we're agreed that the concept of infinite sized sets and the concept of
"points at infinity" are different, and for teaching about "infinity" it might
be useful to separate them explicitly.

Interesting observation. Thanks.

~~~
lmkg
Glad we cleared that up, and glad I could provide some insight =).

Also, thanks for being civil in response to my show-off-y reply.

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gabrielroth
I loved loved loved learning the diagonal proof in college. I've done it for
friends occasionally, as a demonstration of the idea that math can be awesome.

There was one guy in the class who simply wouldn't accept it. He kept saying,
"But so then you add that new number to the list!" I didn't know that he was
representative of a whole larger set of people.

I just posted this to his Facebook page.

~~~
bmm6o
> He kept saying, "But so then you add that new number to the list!"

I think if you get down to it, this reveals an interesting point. I think it
suggests that the logic behind the proof wasn't made clear enough, that it's
not a constructive proof, it's one of the first sophisticated proofs by
contradiction that a student sees. Since you aren't disproving a concrete
thing - you are showing that some hypothetical number doesn't appear on some
hypothetical list - it can seem very unsatisfying.

Also, why is it so ridiculous to try to add a missing number to the list?
There's an intuition that you're fighting with a proof like this: 0: Create an
empty list 1: Find a real number not on your list 2: Add it to the list 3:
goto 1 Obviously this process never terminates, but I think you have to think
pretty hard before that bothers you. After all, what is fundamentally
different if I replace "real" with "integer" or "rational"? (rhetorical)

Finally, students are comfortable enough talking about the "set of reals".
Showing that there's no such thing as the "list of reals" reveals that there
must be some difference between a set and a list - but at the time this proof
is presented, has that distinction been made sufficiently clear?

~~~
gabrielroth
That's a good sketch of some of the complexities involved. I remember that
some months after I learned the proof, your rhetorical question 'What is
fundamentally different if I replace "real" with "integer" or "rational"?'
occurred to me, and I had to spend a few weeks thinking about it before the
answer came to me.

In other words, I got to have an experience of actual mathematical thinking
without learning much higher math. This is why I love the diagonal proof, and
the guy who taught it to me: Paul Lockhart, author of the essay 'A
Mathematician's Lament.' (<http://www.maa.org/devlin/devlin_03_08.html>)

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phaedrus
As a programmer, the thing that bothers me about Cantor's diagonalization (and
the whole concept of Real numbers) is that it presumes an infinite number of
bits of information can be stored in each number. So it's kind of like saying
that you can't enumerate all the values of an infinitely large floating point
number using any arbitrarily large, but finite, integer. Of COURSE you can't.
Which is kind of Cantor's point, but at least from the perspective of a
computer scientist and a mathematical constructivist, it seems a little cheap
because I consider the idea of real numbers themselves to be somewhat
"overpowered", leading to contradictions inherent in the very idea. I'd make
an analogy to the question of can God make a rock so big he can't lift it?
Whether you believe in God or not, you can probably recognize that as a
faintly ridiculous question; I feel real numbers _as a concept_ are ridiculous
in the same way.

~~~
RiderOfGiraffes
Without the reals you can't assume that all limits exist. The reals are the
only totally ordered complete field. As such, whether they're "practical" or
"useful" is irrelevant. They are a mathematical object worthy of study in
their own right.

Calculus is easier with the reals than without. The computing you do is an
approximation to the reals - trying to do numerical analysis without the reals
is horrendous. I've seen so-called "constructivists" attempting it, and I'd
rather work with the reals any day.

~~~
phaedrus
Quote: "Without the reals you can't assume that all limits exist." Oh, I agree
completely. I can't assume that all limits exist.

Modded you up because I agree with your premises, but I disagree with your
conclusion. Both constructivism and "regular" mathematics are systems, but
rather than views computable numbers as approximations of reals, I prefer to
think of real valued calculus as a degenerate case numerical analysis where
delta goes to zero.

Here's the thing: most of mathematics was developed prior to computers, and I
question how much of what's in mainstream mathematics is influenced by the
requirement that ultimately the math had to be tractable to humans without the
aid of a computer.

~~~
jerf
Are you offended at the idea that a mathematical thing called a "real" exists,
or offended at the idea that somebody thinks real numbers actually exist in
the real (no pun intended) universe?

Because nobody thinks real numbers actually exist in a usable manner; whether
the base of the universe is "ultimately real" or "ultimately discrete" is an
open problem, but also irrelevant since we can't get to real values (if they
exist) with infinite precision thanks to the various uncertainty principles.

Ultimately, there really isn't anything to "agree" or "disagree" with when it
comes to the reals; they are just a definition, and everyone who knows what
they are talking about know they don't actually exist. See also the Axiom of
Choice; really there isn't anything to "believe" or "not believe" about it.

(I say "offended" and "believe" but I am not trying to put words into your
mouth. I am not satisfied that either of them describe your tone, but I needed
something. Please do clarify if those are wrong.)

~~~
phaedrus
To answer your first question: Yes. I mean, both. (And no, I'm not offended by
your choice of words.)

Since you've restated the discussion on more neutral terms, allow me to
rephrase my position in your terms: The article is about how people have
trouble swallowing Cantor's argument. Yet clearly Cantor's argument is correct
within a framework that includes the reals. And my position is that the
existence of Cantor's argument leads me to question the framework itself.

You say that there really isn't anything to agree or disagree with, that we
are just talking about definitions. Absolutely, I agree. It's unfortunate that
the only words we have to talk about these things also have extra connotations
that might be more forceful than what we really mean. We're really just
talking about the consequences of manipulating symbols under certain rules.

HOWEVER... Isn't the whole point of math that you expect at some point to
analyze the result of the symbol manipulation and "read off" an interpretable
meaning from the answer? At that point may you not make a judgement that the
combination of the system you choose and your interpretation of it lead to
"believable" results? Is it entirely correct to say that there is no such
thing as "agreeing" or "disagreeing" with a definition of a mathematical
object, if putting it into the system leads to results you don't consider
believable?

~~~
jerf
"Isn't the whole point of math that you expect at some point to analyze the
result of the symbol manipulation and 'read off' an interpretable meaning from
the answer?'

Ah, and therein lies the rub. In math terminology, I think that that _itself_
is an axiom, in the sense that you can choose it or not choose it, and it will
strongly affect the rest of the system. If you accept the axiom, you end up
going down the constructivist path. If you do not, you end up in "conventional
mathematics" (i.e., what you get for a real math degree in college).

Personally, as a programmer I am also sympathetic to the constructivist
viewpoint, but I am happy to settle for everyone involved understanding when
they've left the physical universe (real numbers, axiom of choice, etc) and
when they haven't. If some people want to screw around in the unreal universe,
more power to them, doesn't have to bother me. And they often produce useful
approximations for real-life use. As RiderOfGiraffes said, there are proofs
that work on reals much better than any actual number, and I observe that in
practice the resulting maths work out pretty well in the real world, even if
they aren't actually physical.

(We have _way_ larger problems dealing with floating point, monsters of our
own creation, than we do in dealing with the fact that our equations of motion
are defined based on real numbers that don't actually correspond to anything,
where the inaccuracies are typically beyond our measuring ability and
dominated by general inaccuracies of the input data which are far larger than
the granularity of the universe.)

But I do think maintaining that distinction is definitely important. I further
observe that while I can't speak for all of mathematics, the mathematicians
are very careful to distinguish what takes the Axiom of Choice and what does
not, just to draw one example. They are pretty good about showing their work.

I guess the takeaway is that you certainly aren't alone; many people agree
that real numbers are ridiculous in the way you are sensing. But they can be a
useful fiction.

------
gprisament
This diagonal method is also how you can prove the halting problem is
unsolvable.

------
sophacles
The paper linked in the article is gem.

~~~
lisper
Yes, especially the last sentence. But don't skip ahead. It would be like
listening to the punch line of a joke before hearing the setup.

------
gnosis
Infinity is a process, not a number. And, as such, one infinity can not be
greater than another infinity, as the relationship of being "greater than"
only applies to numbers, not to processes.

~~~
bmm6o
If that's true, you should be able to find a flaw in Cantor's proof. However,
your statement falls into the category of "not even wrong".

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MarkPNeyer
I've always argued that this fact was irrelevant, because although the set of
'all reals' are uncountable, the set of computable numbers IS countable. The
fact that there are numbers which cannot be represented by computers is
interesting but has 0 practical applications.

~~~
sophacles
So let us look at just what you can fit into a 32 bit floating number. I know
you can do reals in other ways, with arbitrary precision libraries, but the
point is still the same. Anyway, the 32bit float. The fact that I have a
limited amount of precision, means that errors exist as a result of truncation
and rounding. For details search google (the main reason for my float
example). The cumulative effects of these can be large enough be represented
within the given amount of precision. Since we now have a proof that those
numbers are innumerably many, we should develop techniques to deal with it,
rather than just add precision.

