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Attempts to Refute Cantor's Uncountability Proof?
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Hatto von Aquitanien
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Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 7:59 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Charlie wrote:


Quote:
It is really quite easy, formulate your mathematics so that ALL sets
are either finite or denumerable. Close to the Greek approach of
rejecting actual infinities.

Or I invoke the axiom of transcendence and don't try to do silly things such
as apply methods defined for finite set to infinite sets where they are
inappropriate.

Quote:
Then Cantor's argument would show that
the real numbers is not a set, but some kind of class, beyond your
range of discussion, safely ignorable.

But if you want to talk about the (completed) reals, then they are
uncountable, so try to get over it.

In my opinion, Cantors argument, which shows up in many forms
in many places, is too beautiful not to be meaningful.

Perhaps, but does it mean what it is commonly understood to mean? To me the
idea that there is a bijective mapping between the natural numbers and the
positive rationales is ...., ...., ????, ..., uh, ....., silly.
--
Nil conscire sibi
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Hatto von Aquitanien
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Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 8:20 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

kunzmilan wrote:

Quote:

We write the rational numbers from the bottom as 1/inf, 2/inf, ...till
inf/inf. This list does not contain the irrational numbers as (sq. root
from 2)/inf, and similarly (sq. root from 2)/n, n =/> 2, since (sq.
root from 2) was not in the original list.
kunzmilan

So your method of generating all the members of an infinite set failed.

Try this: Since I am told that it is meaningful to talk about generation ad
infinitum, why not start by something simple. I have an algorithm which
generates n.0, n.1,..., n.9, on the first pass. On the second, it
generates n.01, n.02, ..., n.09, n.10, n.11, ..., n.19...,n.99. Where n is
the non-negative integer being visited. First visit 0, and make one pass,
then visit 1 and make one pass, return to 0 and make a second pass, then to
1 and make a second pass, then visit two, etc.... Eventually, you will
construct every number representable in decimal notation.
--
Nil conscire sibi
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Stephen Montgomery-Smith1
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Joined: 01 May 2005
Posts: 487

PostPosted: Sat Jul 08, 2006 8:40 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
I'm interested to know what attempts have been made to refute Cantor's proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from which
one might attempt to refute his argument. But before I spend a lot to time
trying to formulate my own argument, it seems reasonable to seek prior art.
Can anybody suggest a source which examines this topic?

I think that the whole concept of infinities upon infinities is
philosophically disturbing. Nevertheless, Cantor's proof in the context
of modern set theory is entirely correct and does provide a proper
formal proof of the uncountability of the reals. On the other hand, if
you reject the infinity upon infinities that modern set theory
postulates, then the truth or falsity of Cantor's result becomes
meaningless.

Nevertheless the logic behind his argument appears in many other places.
For example, you can rewrite the diagonal argument to create a
constructive proof that transcendental numbers exist, or that
non-computable functions exist, or that non-provable statements exist in
number theory (Turing's and Goedel's Theorems respectively). Thus even
if one rejects the infinities upon infinities, Cantor's diagonal
argument still impacts our reality.

I saw in another post that you could not accept the idea that the
rational numbers and the natural numbers can be placed into one to one
correspondence. But there exist effective, explicit ways to enumerate
all ordered pairs. So you are really onto a loser with this one.

Finally another poster talked about attempts to show ZF is inconsistent.
Many years ago I tried this myself. There are two common axiomatic
set theories, ZF and NBG, which loosely speaking are set theory without
classes and set theory with classes. It can be shown that one of these
theories is consistent if and only if the other is. But then I argued
that the class of all sets should be a model of ZF in NBG, thus
providing a proof in NBG that ZF is consistent. By Goedel's Theorem, it
would follow that ZF is inconsistent. It took me quite a while to find
the flaw in this argument.

Stephen
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david petry
science forum Guru


Joined: 18 May 2005
Posts: 503

PostPosted: Sat Jul 08, 2006 8:43 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
I'm interested to know what attempts have been made to refute Cantor's proof
that the real numbers are not denumerable? ...
Can anybody suggest a source which examines this topic?

http://groups.google.com/group/sci.logic/msg/02ee220b035488f9?hl=en&

http://groups.google.com/group/sci.logic/msg/b8cbe85669f24e80?hl=en&

http://groups.google.com/group/sci.math/msg/8245894cf9c14ac6?hl=en&
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Virgil
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Joined: 24 Mar 2005
Posts: 5536

PostPosted: Sat Jul 08, 2006 8:57 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <D7GdnVOPCfRdky3ZnZ2dnUVZ_vydnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:

Quote:
In my opinion, Cantors argument, which shows up in many forms
in many places, is too beautiful not to be meaningful.

Perhaps, but does it mean what it is commonly understood to mean? To me the
idea that there is a bijective mapping between the natural numbers and the
positive rationales is ...., ...., ????, ..., uh, ....., silly.

As I have seen such bijections, and even created a few of my own, I find
Hatto's attitude even sillier.
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Robert Kolker
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Joined: 23 Apr 2005
Posts: 1756

PostPosted: Sat Jul 08, 2006 10:45 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:

Quote:
I'm interested to know what attempts have been made to refute Cantor's proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from which
one might attempt to refute his argument. But before I spend a lot to time
trying to formulate my own argument, it seems reasonable to seek prior art.
Can anybody suggest a source which examines this topic?

The attempts are futile. Cantor's proof is good.

Bob Kolker
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Dave L. Renfro
science forum Guru


Joined: 29 Apr 2005
Posts: 570

PostPosted: Sat Jul 08, 2006 10:45 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote (to kunzmilan):

Quote:
So your method of generating all the members of an infinite
set failed.

Try this: Since I am told that it is meaningful to talk about
generation ad infinitum, why not start by something simple.
I have an algorithm which generates n.0, n.1,..., n.9, on the
first pass. On the second, it generates n.01, n.02, ..., n.09,
n.10, n.11, ..., n.19...,n.99. Where n is the non-negative
integer being visited. First visit 0, and make one pass,
then visit 1 and make one pass, return to 0 and make a second
pass, then to 1 and make a second pass, then visit two, etc....
Eventually, you will construct every number representable in
decimal notation.

At what point in your list will 1/3 be reached? A rough estimate
would be acceptable.

Dave L. Renfro
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 11:16 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Dave L. Renfro wrote:

Quote:
Hatto von Aquitanien wrote (to kunzmilan):

So your method of generating all the members of an infinite
set failed.

Try this: Since I am told that it is meaningful to talk about
generation ad infinitum, why not start by something simple.
I have an algorithm which generates n.0, n.1,..., n.9, on the
first pass. On the second, it generates n.01, n.02, ..., n.09,
n.10, n.11, ..., n.19...,n.99. Where n is the non-negative
integer being visited. First visit 0, and make one pass,
then visit 1 and make one pass, return to 0 and make a second
pass, then to 1 and make a second pass, then visit two, etc....
Eventually, you will construct every number representable in
decimal notation.

At what point in your list will 1/3 be reached? A rough estimate
would be acceptable.

Dave L. Renfro

When I reach countable infinity.
--
Nil conscire sibi
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jt64@tele2.se
science forum beginner


Joined: 05 Jun 2006
Posts: 30

PostPosted: Sat Jul 08, 2006 11:41 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Well let us say Hatto have a logic of his own, but he knows you alot
better than you know him, i have understand that thru the conversation.
You are not a good judge of character even if Hatto would had one.
Gene Ward Smith skrev:

Quote:
Hatto von Aquitanien wrote:
Gene Ward Smith wrote:


You find mathematics as a whole unconvincing,

Yours is among the most moronic statements I have read in this newsgroup,
and you have some tough competition.

You've made dumb remarks (as well as remarks both rude and stupid, like
the above) all too often, so you are not in a good postion to berate
someone else for their alleged idiocy. The above remark is itself
strikingly moronic given that James Harris & co post on this newsgroup.

"I find X to be unconvincing" is something you say over, and over, and
over, and over. I find you to be unconvincing. You don't seem to want
to learn mathematics, just to give yourself a reason to adopt superior
airs by pissing all over it.

Why don't you toddle off and learn something?
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Dave L. Renfro
science forum Guru


Joined: 29 Apr 2005
Posts: 570

PostPosted: Sat Jul 08, 2006 11:49 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote (to kunzmilan):

Quote:
So your method of generating all the members of an infinite
set failed.

Try this: Since I am told that it is meaningful to talk about
generation ad infinitum, why not start by something simple.
I have an algorithm which generates n.0, n.1,..., n.9, on the
first pass. On the second, it generates n.01, n.02, ..., n.09,
n.10, n.11, ..., n.19...,n.99. Where n is the non-negative
integer being visited. First visit 0, and make one pass,
then visit 1 and make one pass, return to 0 and make a second
pass, then to 1 and make a second pass, then visit two, etc....
Eventually, you will construct every number representable in
decimal notation.

Dave L. Renfro wrote:

Quote:
At what point in your list will 1/3 be reached? A rough estimate
would be acceptable.

Hatto von Aquitanien wrote:

Quote:
When I reach countable infinity.

The context is a list that has a first element, a second element,
a third element, and so on, for each positive integer. There is
no "countable infinite" position on such a list.

To show that the positive rationals have the same cardinality
as the positive integers, you need to assign (in a unique way)
a certain positive rational number to '1', a certain positive
rational number to '2', a certain positive rational number
to '3', and so on, in such a manner that every positive
rational number is used up. There is no "countable infinity"
among the numbers '1', '2', '3', etc. -- you have to stay
with the numbers '1', '2', '3', etc.

Besides, even if there were a "countable infinity" among the
positive integers (and there isn't), you'd then have to tell us
what corresponds to the infinite decimal for 1/6, and then what
corresponds to the infinite decimal for 1/9, and for many other
(infinitely many other!) positive rational numbers as well.
Since you've already used up all the numbers '1', '2', '3', ...
and "countable infinity", you can't use any of these again
for 1/6, 1/9, etc.

Dave L. Renfro
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sun Jul 09, 2006 12:11 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Robert J. Kolker wrote:

Quote:
Hatto von Aquitanien wrote:

I'm interested to know what attempts have been made to refute Cantor's
proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from
which
one might attempt to refute his argument. But before I spend a lot to
time trying to formulate my own argument, it seems reasonable to seek
prior art. Can anybody suggest a source which examines this topic?

The attempts are futile. Cantor's proof is good.

Bob Kolker

Cantor's argument is about string manipulation. According to his
assumptions I can manipulate infinite lists of infinitely long strings. I
will only need a countably infinite number of zeros (actually these really
aren't essential, but it the will enhance the user's experience,) and an
equal number of '.'s, so it shouldn't eat up too much paradisal RAM. We
agree that Cantor's second diagonal method doesn't apply to the integers,
right? We append .0 to each of the countably infinite number of integers,
and perform a string reverse on each one.

--
Nil conscire sibi
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matt271829-news@yahoo.co.
science forum Guru


Joined: 11 Sep 2005
Posts: 846

PostPosted: Sun Jul 09, 2006 1:23 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
Robert J. Kolker wrote:

Hatto von Aquitanien wrote:

I'm interested to know what attempts have been made to refute Cantor's
proof
that the real numbers are not denumerable? Quite honestly, I find the
second diagonal method unconvincing. There are a few directions from
which
one might attempt to refute his argument. But before I spend a lot to
time trying to formulate my own argument, it seems reasonable to seek
prior art. Can anybody suggest a source which examines this topic?

The attempts are futile. Cantor's proof is good.

Bob Kolker

Cantor's argument is about string manipulation. According to his
assumptions I can manipulate infinite lists of infinitely long strings. I
will only need a countably infinite number of zeros (actually these really
aren't essential, but it the will enhance the user's experience,) and an
equal number of '.'s, so it shouldn't eat up too much paradisal RAM. We
agree that Cantor's second diagonal method doesn't apply to the integers,
right? We append .0 to each of the countably infinite number of integers,
and perform a string reverse on each one.


This will produce a list of a subset of the rational numbers between
zero and one (specifically, those that have a terminating decimal
representation).
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Dik T. Winter
science forum Guru


Joined: 25 Mar 2005
Posts: 1359

PostPosted: Sun Jul 09, 2006 1:40 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <D7GdnVOPCfRdky3ZnZ2dnUVZ_vydnZ2d@speakeasy.net> Hatto von Aquitanien <abbot@AugiaDives.hre> writes:
....
Quote:
Perhaps, but does it mean what it is commonly understood to mean? To me the
idea that there is a bijective mapping between the natural numbers and the
positive rationales is ...., ...., ????, ..., uh, ....., silly.

That is just a point of view. On the other hand, it is possible to construct
such a bijection. So, what part is silly? Use the following injection
from the positive rationals to the positive naturals:

We use phi(n) (the Euler totient functions that gives the number of positive
integers less than n that are relatively prime to n (note: 1 is coprime to 1
in this context).

Now map a/b, where a and b are coprime to
k(a, b) + sum{i = 1..b-1) phi(i)
so we have still to define k(a, b).

k(a, b) is the number n, such that from 1 onwards to n, a is the n-th number
coprime to b.

Now you may easily prove that the mapping is both surjecive and injective,
and so is a bijection.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
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Dik T. Winter
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Joined: 25 Mar 2005
Posts: 1359

PostPosted: Sun Jul 09, 2006 1:44 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <U8udnUfAH_BqoS3ZnZ2dnUVZ_o2dnZ2d@speakeasy.net> Hatto von Aquitanien <abbot@AugiaDives.hre> writes:
Quote:
Dave L. Renfro wrote:
....
At what point in your list will 1/3 be reached? A rough estimate
would be acceptable.

When I reach countable infinity.

At one step you will reach infinitely many numbers at once? Amazing.
More so, because in all other steps you add only one number to the
set of numbers you have reached.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
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Hatto von Aquitanien
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Joined: 19 Nov 2005
Posts: 410

PostPosted: Sun Jul 09, 2006 2:03 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Dave L. Renfro wrote:

Quote:
Hatto von Aquitanien wrote (to kunzmilan):

So your method of generating all the members of an infinite
set failed.

Try this: Since I am told that it is meaningful to talk about
generation ad infinitum, why not start by something simple.
I have an algorithm which generates n.0, n.1,..., n.9, on the
first pass. On the second, it generates n.01, n.02, ..., n.09,
n.10, n.11, ..., n.19...,n.99. Where n is the non-negative
integer being visited. First visit 0, and make one pass,
then visit 1 and make one pass, return to 0 and make a second
pass, then to 1 and make a second pass, then visit two, etc....
Eventually, you will construct every number representable in
decimal notation.

Dave L. Renfro wrote:

At what point in your list will 1/3 be reached? A rough estimate
would be acceptable.

Hatto von Aquitanien wrote:

When I reach countable infinity.

The context is a list that has a first element, a second element,
a third element, and so on, for each positive integer. There is
no "countable infinite" position on such a list.

Every number I generate increments a counter by 1.

Quote:
To show that the positive rationals have the same cardinality
as the positive integers, you need to assign (in a unique way)
a certain positive rational number to '1', a certain positive
rational number to '2', a certain positive rational number
to '3', and so on, in such a manner that every positive
rational number is used up. There is no "countable infinity"
among the numbers '1', '2', '3', etc. -- you have to stay
with the numbers '1', '2', '3', etc.

And never mind that you are using integers up a lot faster than you are
counting them. That simple fact right there is enough to make the whole
proposition hard to accept. That really is the crux of the argument that
methods created for dealing with finite sets are being abused by applying
them to infinite sets. If I ask at any given point during this enumeration
process what's the difference between the number of integers already
counted, and the number consumed, the latter explodes. One typically
considers such numerical behavior to be divergence. I guess the argument
Cantor will give is that at any given point in the process, there is a
bijective map, and then apply induction.

Quote:
Besides, even if there were a "countable infinity" among the
positive integers (and there isn't), you'd then have to tell us
what corresponds to the infinite decimal for 1/6, and then what
corresponds to the infinite decimal for 1/9, and for many other
(infinitely many other!) positive rational numbers as well.

I could throw those in the loop as well. I apply a predicate which
determines if there is a finite decimal representation for the rational
number, if so I ignore it, if not I add the infinite decimal
representation. That kind of messes up my original goal of saturating the
decimal places sequentially. OTOH, I just accepted an infinite chunk of
data. That deserves closer examination. It smells a whole lot like
circulus in demonstrando.


--
Nil conscire sibi
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