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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 1:38 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Aatu Koskensilta wrote:

Quote:
Hatto von Aquitanien wrote:
Much to my surprise I have discovered there have been many prominent
thinkers who were not persuaded by Cantor's proposition.

Sure. But they weren't idiotic enough to think Cantor's proof was
flawed. Rather, they refused to accept the concept of an arbitrary
infinite set and the hierarchy of transfinite numbers.

quote

url='http://uk.geocities.com/frege%40btinternet.com/cantor/wittgensteinquotes.htm#rfm>

Remarks on the Foundations of Mathematics

V. 7. Imagine set theory's having been invented by a satirist as a kind of
parody on mathematics. ? Later a reasonable meaning was seen in it and it
was incorporated into mathematics. (For if one person can see it as a
paradise of mathematicians, why should not another see it as a joke?) p.
264

(cf Hilbert, D. Uber das Unendliche. Mathematische Annalen 95 (1926 In
Putnam / Benacerraf 183-201, p.191) "No one shall drive us out of the
paradise which Cantror has created for us".

II.15 A clever man got caught in this net of language! So it must be an
interesting net.

II.16 The *mistake* *begins* *when* *one* *says* *that* *the* *cardinal*
*numbers* *can* *be* *ordered* *in* *a* *series* . For what concept has
one of this ordering? One has of course a concept of an infinite series,
but here that gives us at most a vague idea, a guiding light for the
formation of a concept. For the concept itself is abstracted from this and
from other series; or: the expression stands for a certain analogy between
cases, and it can e.g. be used to define provisionally a domain that one
wants to talk about.

That, however, is not to say that the question: "Can the set R be ordered in
a series?" has a clear sense. For this question means e.g.: Can one do
something with these formations, corresponding to the ordering of the
cardinal numbers in a series? Asked: "Can the real numbers be ordered in a
series?" the conscientious answer might be "For the time being I can't form
any precise idea of that". ? "But you can order the roots and the algebraic
numbers for example in a series; so you surely understand the
expression!" ? To put it better, I have got certain analogous formations,
which I call by the common name 'series'. But so far I haven't any certain
bridge from these cases to that of 'all real numbers'. Nor have I any
general method of of trying whether such-and-such a set 'can be ordered in
a series'.

Now I am shewn the diagonal procedure and told: "Now here you have the proof
that this ordering can't be done here". But I can reply "I don't know ? to
repeat ? what it is that can't be done here". Though I can see that you
want to show a difference between the use of "root", "algebraic number",
&c. on the one hand, and "real number" on the other. Such a difference as,
e.g. this: roots are called "real numbers", and so too is the diagonal
number formed from the roots. And similarly for all series of real
numbers. For this reason it makes no sense to talk about a "series of all
real numbers", just because the diagonal number for each series is also
called a "real number". ? Would this not be as if any row of books were
itself ordinarily called a book, and now we said: "It makes no sense to
speak of 'the row of all books', since this row would itself be a book."

II.17. Here it is very useful to imagine the diagonal procedure for the
production of a real number as having been well known before the invention
of set theory, and familiar even to school-children, as indeed might very
well have been the case. For this changes the aspect of Cantor's
discovery. The discovery might very well have consisted merely in the
interpretation of this long familiar elementary calculation.

II.18. For this kind of calculation is itself useful. The question set
would be perhaps to write down a decimal number which is different from the
numbers:

0.1246798 ?
0.3469876 ?
0.0127649 ?
0.3426794 ?
????? (Imagine a long series)

The child thinks to itself: how am I to do this, when I should have to look
at all the numbers at once, to prevent what I write down from being one of
them? Now the method says: Not at all: change the first place of the first
number, the second of the second one &c. &c., and you are sure of having
written down a number that does not coincide with any of the given ones.
The number got in this way might always be called the diagonal number.

II.21 Our *suspicion* ought always to be *aroused* when a *proof* *proves*
*more* *than* *its* *means* *allow* it. Something of this sort might be
called 'a *puffed-up* *proof* '.
</quote>

--
Nil conscire sibi
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Jürgen R.
science forum beginner


Joined: 06 Feb 2006
Posts: 12

PostPosted: Sat Jul 08, 2006 2:04 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

On Sat, 08 Jul 2006 07:09:10 -0400, Hatto von Aquitanien
<abbot@AugiaDives.hre> wrote:

Quote:
Jürgen Ren wrote:

On Sat, 08 Jul 2006 04:06:56 -0400, Hatto von Aquitanien
abbot@AugiaDives.hre> wrote:

I'm interested to know what attempts have been made to refute Cantor's
proof that the real numbers are not denumerable?

It is one of the favorite playground of mathematics quacks.

Such as Frege, Weyl, Witgenstein, Poincarè, Brouwer...?

None of these people, with the possible exception of Wittgenstein, who
was not a mathematician, doubted the validity of the argument. They
did, however, worry about the conceptual basis of some of the
premises.

By quack I mean people like you and Harris and Plutonium.

Obviously, you have been searching for quotes in Wikipedia, which is
not a very clever thing to do. For example, what Weyl meant is
unclear; and if anything Wittgenstein ever meant is to become clear
you will need considerable context.

Quote:

Quite honestly, I find the
second diagonal method unconvincing.

Have you considered that this mayy be a limitation on your part that
has nothing to do with the validity of the argument?

It might be. I find it reassuring, however, that Hermann Weyl appears to
have held a very similar view to mine in this regard. A fact of which I
was unaware when I began this thread.

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?

Yes, but I won't.

As you will.
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Aatu Koskensilta
science forum Guru Wannabe


Joined: 17 May 2005
Posts: 277

PostPosted: Sat Jul 08, 2006 2:21 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
Aatu Koskensilta wrote:

Sure. But they weren't idiotic enough to think Cantor's proof was
flawed. Rather, they refused to accept the concept of an arbitrary
infinite set and the hierarchy of transfinite numbers.

quote
url='http://uk.geocities.com/frege%40btinternet.com/cantor/wittgensteinquotes.htm#rfm

Wittgenstein was notoriously hostile to set theory and its conception of
infinity. It is not clear, however, whether it was the interpretation -
in some sense of the word - of Cantor's results he objected to, or the
proofs themselves. The former would be more in line of his idea of
'philosophy leaving everything as it is'. In any case, even if it was
the proofs themselves, why should we care? Should we be suspicious of
Gödel's proof too, given Wittgenstein's famous misgivings about it?

(Of course, according to a story Wittgenstein understood the proof
perfectly well once it was explained to him by Kreisel and he didn't
rely merely on the introductory remarks in Gödel's paper).

--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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Dave L. Renfro
science forum Guru


Joined: 29 Apr 2005
Posts: 570

PostPosted: Sat Jul 08, 2006 2:43 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?
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?

Extensive excerpts from: Uri Fidelman, "Hemispheric basis for
paradoxes and diagonal processes in mathematics", International
Journal of Mathematical Education in Science and Technology
18 #1 (1987), 61-66.
http://groups.google.com/group/sci.math/msg/22105794d2c59429

William Dilworth, "A correction in set theory", Transactions
of the Wisconsin Academy of Sciences, Arts and Letters 62
(1974), 205-216. [MR 58 #16089]
http://digital.library.wisc.edu/1711.dl/WI.WT1974

Excerpt from Underwood Dudley's 1992 book "Mathematical Cranks"
where Dudley discusses Dilworth's paper.
http://groups.google.com/group/sci.math/msg/bcb253de1b6043fb

Usenet posts about the lawsuit Dilworth filed against Dudley.
One of these posts is by Dudley himself (Nov, 15, 2003),
and Dudley describes the lengths at which Dilworth (now dead,
by the way) tried to attack him through the court system.
http://tinyurl.com/cffrn

Circuit Opinion for Dilworth vs. Dudley. (Dilworth lost.)
http://www.law.emory.edu/7circuit/jan96/95-2282.html

Wilfrid Hodges, "An editor recalls some hopeless papers",
Bulletin of Symbolic Logic 4 (1998), 1-16.
[MR 99c:03007; Zbl 979.03002]
http://www.emis.de/cgi-bin/MATH-item?0979.03002 [Zbl review]
http://www.math.ucla.edu/~asl/bsl/04-toc.htm [.ps file of paper]
http://groups-beta.google.com/group/sci.math/msg/f0ab89956d4591f3
[The above is a text-only copy posted in sci.math by Bill Dubuque.]

Dave L. Renfro
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 08, 2006 2:48 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Jürgen Ren wrote:

Quote:
On Sat, 08 Jul 2006 07:09:10 -0400, Hatto von Aquitanien
abbot@AugiaDives.hre> wrote:

Jürgen Ren wrote:

It is one of the favorite playground of mathematics quacks.

Such as Frege, Weyl, Witgenstein, Poincarè, Brouwer...?

None of these people, with the possible exception of Wittgenstein, who
was not a mathematician,

In this context Wittgenstein's field was at least a relevant as that of
Mathematics.

Quote:
doubted the validity of the argument. They
did, however, worry about the conceptual basis of some of the
premises.

If the premises are not clearly defined, then the validity any
argument "based" upon them cannot be determined.

Quote:
By quack I mean people like you and Harris and Plutonium.

Obviously, you have been searching for quotes in Wikipedia, which is
not a very clever thing to do.

I've looked at more than Wikipedia. I just happened to stumble upon these
names while searching for information on the topic.

Quote:
For example, what Weyl meant is unclear;

Actually, that wasn't what I was looking for when I found that statement.
If I could lay my hands on a copy of Weyl's _Philosophy of Mathematics and
Natural Science_ I would be more than happy to read it. Nonetheless, what
Weyl said seems clear enough to convince me that his objections are
probably similar to my reservations.

Quote:
and if anything Wittgenstein ever meant is to become clear
you will need considerable context.

I believe I provided sufficient context in a previous post on a different
branch of this thread.


--
Nil conscire sibi
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Jürgen R.
science forum beginner


Joined: 06 Feb 2006
Posts: 12

PostPosted: Sat Jul 08, 2006 3:49 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

On Sat, 08 Jul 2006 10:48:06 -0400, Hatto von Aquitanien
<abbot@AugiaDives.hre> wrote:

Quote:
Jürgen Ren wrote:

On Sat, 08 Jul 2006 07:09:10 -0400, Hatto von Aquitanien
abbot@AugiaDives.hre> wrote:

Jürgen Ren wrote:

It is one of the favorite playground of mathematics quacks.

Such as Frege, Weyl, Witgenstein, Poincarè, Brouwer...?

None of these people, with the possible exception of Wittgenstein, who
was not a mathematician,

In this context Wittgenstein's field was at least a relevant as that of
Mathematics.

LOL - so what do you think his field was?

Here is a quote from a letter, W. to Russell 1918, referring to the
manuscript he then called "Logisch-philosophische Abhandlung" :

"I've got the manuscript here with me. I wish I could copy it out for
you... In fact you would not understand it without previous
explanation as it's written in quite short remarks. (This of course
means that nobody will understand it; although I believe it's all as
clear as crystal. But it upsets all our theory of truth, of classes,
of numbers and all the rest.)"

In other words, the hammer came down, but nobody was able to hear it.

W. was a brilliant and seriously wacky man, there is no doubt of that.
However, what he wrote is a muddle of poorly defined terminology. His
students loved this, of course, similar to Heidegger's: They start
talking in tongues, develop a private dialect - young children
sometimes do this too - and thus feel much superior to those who don't
understand them.

Quote:

doubted the validity of the argument. They
did, however, worry about the conceptual basis of some of the
premises.

If the premises are not clearly defined, then the validity any
argument "based" upon them cannot be determined.

By quack I mean people like you and Harris and Plutonium.

Obviously, you have been searching for quotes in Wikipedia, which is
not a very clever thing to do.

I've looked at more than Wikipedia. I just happened to stumble upon these
names while searching for information on the topic.

For example, what Weyl meant is unclear;

Actually, that wasn't what I was looking for when I found that statement.
If I could lay my hands on a copy of Weyl's _Philosophy of Mathematics and
Natural Science_ I would be more than happy to read it. Nonetheless, what
Weyl said seems clear enough to convince me that his objections are
probably similar to my reservations.

The "statement" is, so far as I can tell, not a quote from Weyl. The
bit about "the Fall and Original Sin of set theory" is Weyl quoting
Brouwer (in Philos. of Math. and the Natural Sciences). The rest of
the quote doesn't occur in the same context and I can't locate it.

When you take Wikipedia as your source you have only yourself to
blame.

Quote:

and if anything Wittgenstein ever meant is to become clear
you will need considerable context.

I believe I provided sufficient context in a previous post on a different
branch of this thread.

No you didn't.
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kunzmilan@atlas.cz
science forum beginner


Joined: 21 Feb 2006
Posts: 42

PostPosted: Sat Jul 08, 2006 6:06 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with 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
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Sat Jul 08, 2006 6:20 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Jürgen Ren wrote:

Quote:
W. was a brilliant and seriously wacky man, there is no doubt of that.
However, what he wrote is a muddle of poorly defined terminology. His
students loved this, of course, similar to Heidegger's: They start
talking in tongues, develop a private dialect - young children
sometimes do this too - and thus feel much superior to those who don't
understand them.

Funny, since he denied that a private language was possible.
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Sat Jul 08, 2006 6:30 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Dave L. Renfro wrote:

Quote:
Circuit Opinion for Dilworth vs. Dudley. (Dilworth lost.)
http://www.law.emory.edu/7circuit/jan96/95-2282.html

In it I found this:

Among the terms or epithets that have been held (all in
the cases we've cited) to be incapable of defaming because
they are mere hyperbole rather than falsifiable assertions
of discreditable fact are "scab," "traitor," "amoral," "scam,"
"fake," "phony," "a snake-oil job," "he's dealing with half
a deck," and "lazy, stupid, crap-shooting, chicken-stealing
idiot."

I think this is a pretty good place to turn if we need to expand our
vocabulary here on sci.math. Which posters are lazy, stupid,
crap-shooting, chicken-stealing idiots (not defamatory, so don't bother
trying to sue) and which are merely dealing with half a deck?
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fishfry
science forum Guru Wannabe


Joined: 29 Apr 2005
Posts: 299

PostPosted: Sat Jul 08, 2006 6:32 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <0vednYwW3edP-jLZnZ2dnUVZ_r2dnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> 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?

Only among the cranks.
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Sat Jul 08, 2006 6:52 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <WP2dnReap8F_6jLZnZ2dnUVZ_vSdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:


Quote:
"...classical logic was abstracted from the mathematics of finite sets and
their subsets...Forgetful of this limited origin, one afterwards mistook
that logic for something above and prior to all mathematics, and finally
applied it, without justification, to the mathematics of infinite sets.
This is the Fall and original sin of [Cantor's] set theory ..." (Weyl,
1946)

And of what God, is Weyl the prophet?
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Sat Jul 08, 2006 6:55 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <LeCdnU7u6ofo4TLZnZ2dnUVZ_t6dnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:

Quote:
Gene Ward Smith wrote:


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.

You're not very good at the basic logic of sets, I see.

While Hatto may not have read any of JSH's more moronic statements, he
has certainly read some of TO's and RE's.

Thus Hatto reveals that he is himself not very good at judging levels of
moronic logic.
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Sat Jul 08, 2006 6:58 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <ZfudnVOTvJMuMTLZnZ2dnUVZ_rOdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:

Quote:
Much to my surprise I have discovered there have been many prominent
thinkers who were not persuaded by Cantor's proposition.

"For if one person can see it as a paradise of mathematicians, why should
not another see it as a joke?" - Ludwig Wittgenstein

I have heard that many prophets are not honored, particularly in their
own country.
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Charlie
science forum beginner


Joined: 30 Apr 2005
Posts: 24

PostPosted: Sat Jul 08, 2006 7:01 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?
--
Nil conscire sibi

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. 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.
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Sat Jul 08, 2006 7:01 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <Ps-dnXBxMNZEWDLZnZ2dnUVZ_rudnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:

Quote:
I believe I provided sufficient context in a previous post on a different
branch of this thread.

What Hatto believes is not evidence.
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