Attempts to Refute Cantor's Uncountability Proof?
Hatto von Aquitanien
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
Gene Ward Smith
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.
You find mathematics as a whole unconvincing, so I find your lack of
conviction to be unconvincing.
Patrick
Hatto von Aquitanien
> 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.
> so I find your lack of conviction to be unconvincing.
Sorry if I don't accept the holy Aleph-Omega of Mathematics upon the dogma
of others.
--
Nil conscire sibi
Hatto von Aquitanien
I was thinking more in terms of Frege. But I suspect Frege would not have
taking the approach I am likely to take.
"...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)
--
Nil conscire sibi
Gene Ward Smith
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?
Hatto von Aquitanien
>
> 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.
--
Nil conscire sibi
Gene Ward Smith
Hatto von Aquitanien wrote:
> You're not very good at the basic logic of sets, I see.
How the hell would you know? I at least took a course in the subject,
which I doubt you ever did.
Hatto von Aquitanien
>
> Hatto von Aquitanien wrote:
>
>> You're not very good at the basic logic of sets, I see.
>
> How the hell would you know?
Quod erat demonstrandum.
> I at least took a course in the subject, which I doubt you ever did.
Consider asking for a refund.
--
Nil conscire sibi
Jürgen Ren
<ab...@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.
> 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?
> 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.
Hatto von Aquitanien
> On Sat, 08 Jul 2006 04:06:56 -0400, Hatto von Aquitanien
> <ab...@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, Poincarč, Brouwer...?
>> 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.
--
Nil conscire sibi
David C. Ullrich
<ab...@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? Quite honestly, I find the
>second diagonal method unconvincing. There are a few directions from which
>one might attempt to refute his argument.
Guffaw.
> 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 only such source I'm aware of is sci.math. The refutations you
find here make exactly as much sense as someone "refuting" the
construction of the reals from the rationals via dedekind cuts
by pointing out that pi is not rational. Ie, they exhibit a
basic misunderstanding of the argument that they puport to refute.
************************
David C. Ullrich
Aatu Koskensilta
The topic occasionally crops up in sci.logic and sci.math, as well as
some other newsgroups. Going trough the archives you'll find any number
of inspiring refutations of Cantor's diagonal proof. Proving that ZFC is
inconsistent is another favourite pastime you could try your hand at.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Hatto von Aquitanien
> 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 topic occasionally crops up in sci.logic and sci.math, as well as
> some other newsgroups. Going trough the archives you'll find any number
> of inspiring refutations of Cantor's diagonal proof. Proving that ZFC is
> inconsistent is another favourite pastime you could try your hand at.
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
--
Nil conscire sibi
Aatu Koskensilta
> 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.
Hatto von Aquitanien
> 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
Jürgen Ren
<ab...@AugiaDives.hre> wrote:
>Jürgen Ren wrote:
>
>> On Sat, 08 Jul 2006 04:06:56 -0400, Hatto von Aquitanien
>> <ab...@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, Poincarč, 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.
Aatu Koskensilta
> 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).
Dave L. Renfro
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
Hatto von Aquitanien
> On Sat, 08 Jul 2006 07:09:10 -0400, Hatto von Aquitanien
> <ab...@AugiaDives.hre> wrote:
>
>>Jürgen Ren wrote:
>>> It is one of the favorite playground of mathematics quacks.
>>
>>Such as Frege, Weyl, Witgenstein, Poincarč, 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.
> 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.
> 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
Jürgen Ren
<ab...@AugiaDives.hre> wrote:
>Jürgen Ren wrote:
>
>> On Sat, 08 Jul 2006 07:09:10 -0400, Hatto von Aquitanien
>> <ab...@AugiaDives.hre> wrote:
>>
>>>Jürgen Ren wrote:
>
>>>> It is one of the favorite playground of mathematics quacks.
>>>
>>>Such as Frege, Weyl, Witgenstein, Poincarč, 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.
>
>> 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.
>
>> 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.
kunzmilan
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
Gene Ward Smith
Jürgen Ren wrote:
> 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.
Gene Ward Smith
Dave L. Renfro wrote:
> 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?
fishfry
Hatto von Aquitanien <ab...@AugiaDives.hre> wrote:
Only among the cranks.
Virgil
Hatto von Aquitanien <ab...@AugiaDives.hre> wrote:
> "...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?
Virgil
Hatto von Aquitanien <ab...@AugiaDives.hre> wrote:
> 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.
Virgil
Hatto von Aquitanien <ab...@AugiaDives.hre> wrote:
> 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.
Charlie
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?
> 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.
Virgil
Hatto von Aquitanien <ab...@AugiaDives.hre> wrote:
> I believe I provided sufficient context in a previous post on a different
> branch of this thread.
What Hatto believes is not evidence.
Hatto von Aquitanien
> 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.
> 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
Hatto von Aquitanien
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
Stephen Montgomery-Smith
> 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
david petry
Hatto von Aquitanien wrote:
> I'm interested to know what attempts have been made to refute Cantor's proof
> 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&
Virgil
Hatto von Aquitanien <ab...@AugiaDives.hre> wrote:
> > 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.
Robert J. Kolker
> 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
Dave L. Renfro
> 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
Hatto von Aquitanien
When I reach countable infinity.
--
Nil conscire sibi
jt...@tele2.se
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:
> 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?
Dave L. Renfro
>>> 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.
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
Hatto von Aquitanien
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
matt271...@yahoo.co.uk
This will produce a list of a subset of the rational numbers between
zero and one (specifically, those that have a terminating decimal
representation).
Dik T. Winter
> Dave L. Renfro wrote:
...
> > At what point in your list will 1/3 be reached? A rough estimate
> > would be acceptable.
>
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/
Dik T. Winter
...
> 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.
Hatto von Aquitanien
> 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.
> 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.
> 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
Hatto von Aquitanien
>
> Hatto von Aquitanien wrote:
It also maps bijectively to "completed infinity". There seems to be a
problem here. For every rational number we previously claimed there was a
corresponding integer. There are rational numbers which lie between each
pair of consecutive integers ad infinitum, and there are an infinite number
of such rational numbers between each pair of integers. Nonetheless, we
did not even succeed in covering the rational numbers between zero and 1.
--
Nil conscire sibi
Hatto von Aquitanien
> In article <D7GdnVOPCfRdky3Z...@speakeasy.net> Hatto von
> Aquitanien <ab...@AugiaDives.hre> writes: ...
> > 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.
With finite sets Dirichlet's pigeons (appear to) keep people honest.
--
Nil conscire sibi
Hatto von Aquitanien
> In article <U8udnUfAH_BqoS3Z...@speakeasy.net> Hatto von
> Aquitanien <ab...@AugiaDives.hre> writes:
> > 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.
I hope you detected the intended sarcasm.
--
Nil conscire sibi
Dave L. Renfro
>>>> 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.
Dave L. Renfro wrote:
>> 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.
Hatto von Aquitanien wrote:
> Every number I generate increments a counter by 1.
Fine. This means the counter will show '1', '2', '3', etc.
But at no point will the counter show "countable infinite".
Dave L. Renfro wrote:
>> 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.
Hatto von Aquitanien wrote:
> 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.
What about the fact that I can list (in the manner we're
talking about) all the positive integers using the positive
even integers? In what follows, the counting is being done
by the positive even integers (left of the arrows) and the
numbers being counted are the positive integers to the
right of the arrows.
2-->1 4-->2 6-->3 8-->4 10-->5 12-->6 14-->7 . . .
In this situation, I'm using up the even integers that I'm
"counting with" much faster than the even integers that are
among the numbers I'm counting. The (arithmetic) difference
between the number of even integers counted vs. the number
of even integers used up increases without bound (or "explodes",
as you say).
Instead of even integers, if I use powers of 2 to count with,
I can make the gaps increase fast enough so that we actually
get the *ratios* between the number of those integers that I'm
using to count with (i.e. the powers of 2) vs. the number of
powers of 2 that get used up to increase without bound:
2-->1 4-->2 8-->3 16-->4 32-->5 64-->6 128-->7 . . .
These are 1-1 correspondences, by the way. In the first case,
each positive even integer n is matched with exactly one positive
integer, namely n/2, and each positive integer m is matched with
exactly one positive even integer, namely 2m. In the second case,
each positive integer power of 2 is matched with exactly one
positive integer, namely the exponent of 2 associated with the
power of 2, and each positive integer m is matched with exactly
one positive integer power of 2, namely 2^m.
With the right approach, one can use the positive integers
"to count" the positive rational numbers. Although Cantor
is credited with this, I don't think it is considered all that
significant, and in fact I believe that Dedekind (who Cantor
was in correspondence with during 1873-74 when Cantor was
taking his first steps into this area) independently came
up with such an approach. Cantor's importance comes from
the fact that: (1) he first considered in a systematic way
what the consequences might be for comparing infinite sets
in this way (by 1-1 correspondences); (2) he managed
to come up with a proof that there exists an infinite set
that can't be "counted" by using the positive integers
(namely, the real numbers); (3) he created an elaborate
set of mathematical tools that proved very useful for
solving problems, and establishing connections, in a wide
variety of mathematical fields.
Dave L. Renfro
Hatto von Aquitanien
> These are 1-1 correspondences, by the way. In the first case,
> each positive even integer n is matched with exactly one positive
> integer, namely n/2, and each positive integer m is matched with
> exactly one positive even integer, namely 2m. In the second case,
> each positive integer power of 2 is matched with exactly one
> positive integer, namely the exponent of 2 associated with the
> power of 2, and each positive integer m is matched with exactly
> one positive integer power of 2, namely 2^m.
Understood. Some of this is actually coming back to me from the days of my
misspent youth. As usual, I really didn't plan on doing more than getting
a sense of who, in terms of serious mathematicians, may have attempted a
refutation of Cantor. In stead I ended up digging in to the topic.
> With the right approach, one can use the positive integers
> "to count" the positive rational numbers. Although Cantor
> is credited with this, I don't think it is considered all that
> significant, and in fact I believe that Dedekind (who Cantor
> was in correspondence with during 1873-74 when Cantor was
> taking his first steps into this area) independently came
> up with such an approach. Cantor's importance comes from
> the fact that: (1) he first considered in a systematic way
> what the consequences might be for comparing infinite sets
> in this way (by 1-1 correspondences); (2) he managed
> to come up with a proof that there exists an infinite set
> that can't be "counted" by using the positive integers
> (namely, the real numbers); (3) he created an elaborate
> set of mathematical tools that proved very useful for
> solving problems, and establishing connections, in a wide
> variety of mathematical fields.
This last point seems somewhat unclear to me. His ideas are certainly
popular, and, right or wrong (sorry, I'm still skeptical[*]) his ideas of
transfinity are interesting. There seems to be some difference of opinion
regarding how essential his contributions are to mathematics. Some
mathematicians seem to perceive them as essential while other seem(ed) to
believe they are superfluous, or even harmful. For myself, what I know to
be attributed to Cantor has mostly provided ground for philosophical
reflection, mental exercises and entertainment. Nothing wrong with any of
these, but I can't think of a lot that has been directly useful in solving
problems.
[*] And there's always "it's true because we define it to be true".
--
Nil conscire sibi
Jonathan Hoyle
> I'm interested to know what attempts have been made to refute Cantor's proof
> that the real numbers are not denumerable?
Quite an impressive collection of such writings can be found at
http://www.crank.net/cantor.html . Such writers, such as yourself,
ignore the fact that Cantor's theorem is proven, and act as if a more
"persuasive" argument is given, that his proof is somehow unproven.
Cranks are an amusing lot, and you will find them appearing here on
sci.math from time to time. One such long standing "Anti-Cantor" crank
is Mike Deeth, who pretends to be an unaging 11 year old named Nathan.
(He assumes this identity presumably because his arguments are so weak
that a 12 year old could dismantle them.) You will find many others on
the crank.net web site.
Hatto von Aquitanien
Some people will believe in the most bizarre absurdities:
http://www.whatreallyhappened.com/wtc7.html
--
Nil conscire sibi
matt271...@yahoo.co.uk
> matt271...@yahoo.co.uk wrote:
>
> >
> > Hatto von Aquitanien wrote:
>
> >> 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).
>
> It also maps bijectively to "completed infinity". There seems to be a
> problem here. For every rational number we previously claimed there was a
> corresponding integer.
That's right. There are various systematic ways to list all the
rational numbers. I recall there's a quite simple iterative way to do
it so that each rational number appears exactly once, in its simplest
form, though I can't immediately find a reference. Thus the rational
numbers can be put into a one-to-one correspondence with the natural
numbers, and are countable (unlike the irrationals).
> There are rational numbers which lie between each
> pair of consecutive integers ad infinitum, and there are an infinite number
> of such rational numbers between each pair of integers. Nonetheless, we
> did not even succeed in covering the rational numbers between zero and 1.
It is also possible (very easy, in fact) to produce an infinite list of
distinct rational numbers that is nevertheless incomplete. This is
possible because an infinite set can be put into one-to-one
correspondence with a subset of itself.
matt271...@yahoo.co.uk
matt271...@yahoo.co.uk wrote:
[snip]
> I recall there's a quite simple iterative way to do
> it so that each rational number appears exactly once, in its simplest
> form, though I can't immediately find a reference.
One that doesn't require you to actually do the reduction to simplest
form "manually" for each fraction, I mean...
Dave L. Renfro
>> [...] he [Cantor] created an elaborate set of mathematical
>> tools that proved very useful for solving problems, and
>> establishing connections, in a wide variety of mathematical
>> fields.
Hatto von Aquitanien wrote (in part):
> This last point seems somewhat unclear to me. His ideas are
> certainly popular, and, right or wrong (sorry, I'm still
> skeptical[*]) his ideas of transfinity are interesting. There
> seems to be some difference of opinion regarding how essential
> his contributions are to mathematics. Some mathematicians seem
> to perceive them as essential while other seem(ed) to believe
> they are superfluous, or even harmful. For myself, what I know
> to be attributed to Cantor has mostly provided ground for
> philosophical reflection, mental exercises and entertainment.
> Nothing wrong with any of these, but I can't think of a lot
> that has been directly useful in solving problems.
>
> [*] And there's always "it's true because we define it to be
> true".
I suspect the "superfluous, or even harmful" examples, for the
most part, fall into one of the following categories:
1. Early mathematicians (writings before about 1910), whose
opinion was based on the applications then available.
2. Older early writers (writings before about 1930), whose
opinion was due to their lack of background in newly
emerging fields and the usual age-related reluctance
to deviate very much from what one learned when younger.
3. Philosophers who either don't understand the concepts
well enough or who take issue with certain philosophical
matters that are _suggested_ by (as opposed to "implied by")
Cantor's work. The latter type often have very little
relevance to actual mathematical work.
4. Mathematicians such as Brouwer who disagreed with certain
aspects of what the underlying rules for mathematical
investigation should be.
5. Mathematicians who are comparing Cantor's work to the top
handful of mathematical advances in human history.
6. Mathematicians who are misquoted or quoted out of context.
Cantor's first paper in set theory (1874) was primarily a
proof for the existence of transcendental numbers. The proof
that the set of real numbers is not countable was the tool.
Cantor's classification of the various derived sets of sets
of real numbers was influential with several researchers
in the 1880's who were studying and extending the results
of Riemann's 1854 paper on trigonometric expansions and what
is now called the Riemann integral (this 1854 work of Riemann's
wasn't generally known until its republication, after Riemann's
death, in 1867), such as Hankel, Dini, Bois-Reymond, Harnack,
and others. Dini's influential 1878 monograph makes extensive
use of these ideas.
Borel, Legesgue, Young, Baire, and others at the beginning of
the 20'th century made extensive use of Cantor's work in their
researches in real and complex analysis. Transfinite induction
and transfinite construction processes were extensively used.
The fact that the collection of Lebesgue measurable sets
[the collection of Lebesgue measurable functions] is a proper
superset of the collection of Borel sets [the collection of
Baire functions] follows from an easy cardinality argument
along with Lebesgue's 1905 [Baire's 1899] transfinite
construction of the Borel sets [Baire functions].
Denjoy's extension of the Lebesgue integral, around 1914-16,
was directly based on a transfinite construction process.
There are many more examples, even if one restricts themselves
to before 1915. After the mid 1920's, when the Polish and
Russian analysis and topology schools began to become very
productive, the applications really took off.
Dave L. Renfro
Lasse
> I'm interested to know what attempts have been made to refute Cantor's proof
> 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?
No. It is impossible to 'refute' Cantor's proof, since it is correct.
If you want to convince yourself of this, you should go through the
proof, and see that indeed each step is justified by the basic axioms
of set theory.
Of course, nothing stops you from choosing to not believe in some of
these axioms, and in fact the "existence" of the real numbers
themselves. However, that is not a "refutation" of Cantor's proof.
I wonder whether you actually know Cantor's proof properly. In fact, it
is a special case of the following, more general fact:
THEOREM. Let X be a set. Then there is no surjective function from X to
its power set (that is, the set of all subsets of X).
PROOF. Let f be a function from X to its power set, and consider the
set
A = { x\in X: x is not an element of f(x) }.
We claim that f(x) is not equal to A for every element of X. (Hence f
is not surjective.)
Indeed, let x be any element of X. We consider two cases:
a) x is an element of f(x). Then x is not an element of A. So A is not
equal to f(x).
b) x is not an element of f(x). Then x is an element of A. So again, A
is not equal to f(x).
QED.
Is this OK with you? If so, then we just need to notice that
a) the power set of the natural number is in one-to-one correspondence
with the set of all sequences of zeros and ones;
b) there is an injective function from the latter set into the real
numbers.
Both of these are also easily explained. So if you have a problem with
the proof, you should say which of these steps you do not understand.
I hope this helps,
Lasse
Jonathan Hoyle
>
> http://www.whatreallyhappened.com/wtc7.html
True, but I'm not sure what WTC cranks have in common with Mathematical
cranks. One can be a crank in one area without being a crank in
another.
Stephen Montgomery-Smith
> 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
david petry
Stephen Montgomery-Smith wrote:
> ... if you reject the infinity upon infinities that modern set theory
> postulates, then the truth or falsity of Cantor's result becomes
> meaningless.
Well said. I believe that is the point most of the "anti-Cantorians"
are trying, but failing, to make.
> 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,
That's not really true. In order to give a constructive proof of the
existence of transcendental numbers, you must show that the digits of
algebraic numbers can actually be computed (and hence, there are bounds
on how closely such numbers can be approximated by rationals), but once
you have shown that, you don't really need the diagonal argument to
show the existence of real numbers that are not algebraic.
Gene Ward Smith
david petry wrote:
> That's not really true. In order to give a constructive proof of the
> existence of transcendental numbers, you must show that the digits of
> algebraic numbers can actually be computed...
But that step is trivial.
Hatto von Aquitanien
> I suspect the "superfluous, or even harmful" examples, for the
> most part, fall into one of the following categories:
[...]
> 2. Older early writers (writings before about 1930), whose
> opinion was due to their lack of background in newly
> emerging fields and the usual age-related reluctance
> to deviate very much from what one learned when younger.
[...]
> 4. Mathematicians such as Brouwer who disagreed with certain
> aspects of what the underlying rules for mathematical
> investigation should be.
[...]
> 6. Mathematicians who are misquoted or quoted out of context.
Of your original list, the above appear to be the items of criticism to
which Weyl, or those citing him, would be most susceptible - though 4 is
more of an observation than a criticism. Since I don't have immediate
access to Weyl's later writings on this matter and am, therefore, at the
mercy of the whims of the Internet community, it is difficult for me to
assess what his opinions actually were. And that assumes that his opinions
were consistent from day to day, and year to year. But with all those
qualifications stated, the impression I have is that he did not accept
Cantor's findings. I know of no mathematician who would have stood in
better stead to judge the matter rightly. That doesn't mean he was
incapable of error. But it does mean his opinions were informed and
contemplated.
> Cantor's first paper in set theory (1874) was primarily a
> proof for the existence of transcendental numbers. The proof
> that the set of real numbers is not countable was the tool.
I don't claim expertise in this area, but I thought it had, by that date,
long been established that some numbers were both irrational and
non-algebraic.
> Denjoy's extension of the Lebesgue integral, around 1914-16,
> was directly based on a transfinite construction process.
>
> There are many more examples, even if one restricts themselves
> to before 1915. After the mid 1920's, when the Polish and
> Russian analysis and topology schools began to become very
> productive, the applications really took off.
I've not encountered these in any applied mathematics. I have a high regard
for what I consider Pure Mathematics - hence my conservativism and pedantry
regarding the use of the term "Axiom". Nonetheless, Kant's warning against
building conclusion upon improper foundations seems apropos here. Since it
appears that many mathematicians are using Cantor's conclusions directly or
indirectly as a foundation upon which to build, the matter of soundness
seems vitally important.
--
Nil conscire sibi
Gene Ward Smith
Hatto von Aquitanien wrote:
> I've not encountered these in any applied mathematics. I have a high regard
> for what I consider Pure Mathematics - hence my conservativism and pedantry
> regarding the use of the term "Axiom".
You do not.
Stephen Montgomery-Smith
> Dave L. Renfro wrote:
>>Cantor's first paper in set theory (1874) was primarily a
>>proof for the existence of transcendental numbers. The proof
>>that the set of real numbers is not countable was the tool.
>
> I don't claim expertise in this area, but I thought it had, by that date,
> long been established that some numbers were both irrational and
> non-algebraic.
I'm not totally sure, but I think that the existence of non-algebraic
integers was proved only very shortly before this date - it was shown
that sum 10^{-n!} is non-algebraic. I think that pi and e are
non-algebraic came later.
>
>
>>Denjoy's extension of the Lebesgue integral, around 1914-16,
>>was directly based on a transfinite construction process.
>>
>>There are many more examples, even if one restricts themselves
>>to before 1915. After the mid 1920's, when the Polish and
>>Russian analysis and topology schools began to become very
>>productive, the applications really took off.
>
>
> I've not encountered these in any applied mathematics. I have a high regard
> for what I consider Pure Mathematics - hence my conservativism and pedantry
> regarding the use of the term "Axiom". Nonetheless, Kant's warning against
> building conclusion upon improper foundations seems apropos here. Since it
> appears that many mathematicians are using Cantor's conclusions directly or
> indirectly as a foundation upon which to build, the matter of soundness
> seems vitally important.
My personal viewpoint is that current mathematics is built upon improper
foundations, but that it is pointless and worthless trying to find the
flaws at this point in history. For example, I believe that Einstein
could never have figured out general relativity had not Riemann and
Gauss figured out the mathematics behind curved space, and as such our
attempts to figure out or correct the flaws in our current models are
just as futile as any attempt to figure out general relativity without
the notions of curved space would have been.
While I believe that the current foundations are deeply flawed (and let
me reiterate - this belief is not based upon scientific reasoning), I do
think that our current foundations are (a) very good and (b) good enough
for now. When the flaws are found, I believe that most of mathematics
will survive - perhaps only the really weird stuff like inaccessible
ordinals will die. Thus it makes very great sense for us to continue
our studies in math as if everything is just fine, and it would be an
exercise in futility to try to get our foundations completely correct.
From your frequent statements "I am not an expert in XXX" I suggest
that you read a good book on mathematical logic. The book from which I
learned this stuff is by Mendolsen (spelling?), and it seems to have so
many editions that it must be considered by others as well to be a good
book. I think that when you know the subject well, you will better
appreciate both the strengths and weaknesses of the modern approach.
Stephen
Hatto von Aquitanien
> 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?
>
> 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.
I have a problem with the idea of assuming that a 1-1 mapping between
rational numbers and the natural numbers has been established. That is, the
step between the first diagonal construction, and the application of the
second diagonal method. It assumes that it is meaningful to contemplate
the result of executing an infinite number of constructive steps. Notice
that when I posted my algorithm for the iterative construction of all
numbers expressed in decimal notation I was asked when I would get to 1/3.
My response was, 'when I reach countable infinity'. That was rejected as
meaningless. Nonetheless, that seems to be exactly what we have to assume
when we apply Cantor's second diagonal method.
So I ask, if Cantor can present his constructive method and then have us
contemplate the result of its completion, why can I not do the same?
> 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.
Or, if I flat-out assert that it is a logical error to assume the completion
of an iterative Dedekind mapping, it Cantor's proof becomes an error. Of
course, taking pot-shots at infinite iteration, in genera, is a dangerous
game. Riemann sums, and many other useful notions become vulnerable.
> 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.
The definition of infinity, "An infinite set is equipollent to one of its
proper subsets", is due to Dedekind.
> 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.
I don't believe you fully appreciated what I was getting at. Suppose you
have a strip of cloth with buttons along one side and buttonholes along the
other.
IIIIII
000000
Label the buttonholes consecutively using the natural numbers. Label the
buttons with the consecutive results of Cantor's first diagonal algorithm.
After some finite number of iterations, identify each button that has an
integer value assigned to it, and button it to the correspondingly labeled
button hole. Notice that the number of unmatched buttons continues to
increase per integer button. Each integer button will add on more
unmatched button than its predecessor. In the realm of finite sets you
will run out of buttons long before you run out of button holes. In
infinite sets you never encounter that problem because the game never ends.
If you are talking about infinite sets, and using a Dedekind recursive
mapping, it is an error to draw a conclusion by contemplating the result of
having completed the mapping.
> 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.
I believe the former is sometimes referred to as Naive Set Theory.
> 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.
I don't follow your reasoning here. Why should proof of the consistency of
ZF as a result of NBG prove ZF is inconsistent by Gödel's Theorem? The
informal formulation of Gödel's result which I remember is that 'no
sufficiently powerful system can be shown to be both consistent and
complete'.
--
Nil conscire sibi
Stephen Montgomery-Smith
First, I am unfamiliar with the terms "first diagonal method" and
"second diagonal method", but I do know that there are several ways to
place the natural numbers in 1 to 1 correspondence with the ordered
pairs. Any one of these methods succeeds in doing the following. Given
an ordered pair I can find a natural number in the correspondence, and
given a natual number I can find the ordered pair it corresponds to. It
might require contemplating the infinite - although some clever choices
of correspondence will even admit an explicit formula. But this is very
different to what you were doing with 1/3. They asked you what natural
number corresponds to 1/3 in your correspondence. You replied
"countable infinity." But "countable infinity" is not a natural number.
Thus 1/3 does not coorespond to anything in your correspondence.
In other enumeration is a list in which you know that anything in your
list will eventually appear after going a finite amount along the list.
You might not know a priori how far along the list you have to go
before you find an item you are looking for, but you do know that if you
keep going, you will get to it.
Thus saying you get to 1/3 after countable infinity does not count. For
it to count, you have to know that 1/3 will appear at some finite place
along the list. It might be unusually large, but it still has to be finite.
>
>
>>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.
>
>
> Or, if I flat-out assert that it is a logical error to assume the completion
> of an iterative Dedekind mapping, it Cantor's proof becomes an error. Of
> course, taking pot-shots at infinite iteration, in genera, is a dangerous
> game. Riemann sums, and many other useful notions become vulnerable.
I'm not familiar with the term "Dedekind mapping."
I disagree. No matter how many buttonholes you go along, you will
always find a button that matches it, even if you have to go for a lot
longer. And vice versa.
>
>
>>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.
>
>
> I believe the former is sometimes referred to as Naive Set Theory.
>
>
>>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.
>
>
> I don't follow your reasoning here. Why should proof of the consistency of
> ZF as a result of NBG prove ZF is inconsistent by Gödel's Theorem? The
> informal formulation of Gödel's result which I remember is that 'no
> sufficiently powerful system can be shown to be both consistent and
> complete'.
>
What it says is that no sufficiently powerful system can be shown to be
consistent and complete within that system, if that system is
consistent. For example, it is easy to show Peano's axioms are
consistent using ZF.
Hatto von Aquitanien
It becomes more clear with each of your posts exactly why you took offense
to my comment about proof by handwaving and/or insult.
--
Nil conscire sibi
Gerry Myerson
Stephen Montgomery-Smith <ste...@math.missouri.edu> wrote:
> Hatto von Aquitanien wrote:
> > Dave L. Renfro wrote:
>
> >>Cantor's first paper in set theory (1874) was primarily a
> >>proof for the existence of transcendental numbers. The proof
> >>that the set of real numbers is not countable was the tool.
> >
> > I don't claim expertise in this area, but I thought it had, by that date,
> > long been established that some numbers were both irrational and
> > non-algebraic.
>
> I'm not totally sure, but I think that the existence of non-algebraic
> integers was proved only very shortly before this date - it was shown
> that sum 10^{-n!} is non-algebraic. I think that pi and e are
> non-algebraic came later.
Liouville did sum 10^{-n!} in 1850, give or take a few years.
Transcendence of e was Hermite, 1873.
Transcendence of pi was Lindemann, 1882.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Hatto von Aquitanien
First diagonal method is the process of visiting each rational number
sequentially and assigning a natural number to it.
http://www.homeschoolmath.net/teaching/rational-numbers-countable.php
The second diagonal method is the generation of a unique sequence from an
infinite ordered list of sequences by visiting the ii-th element of the
sequence and adding something to the new sequence which differs from the
element being visited.
http://uk.geocities.com/fr...@btinternet.com/cantor/diagarg.htm
> but I do know that there are several ways to
> place the natural numbers in 1 to 1 correspondence with the ordered
> pairs. Any one of these methods succeeds in doing the following. Given
> an ordered pair I can find a [unique] natural number in the
> correspondence, and given a natual number I can find the ordered [unique]
> pair it corresponds to. It might require contemplating the infinite -
> although some clever choices of correspondence will even admit an explicit
> formula.
This looks to be worth contemplating. It seems to challenge my notion that
Cantor relied on the completion of an infinite construction.
> But this is very different to what you were doing with 1/3. They asked
> you what natural number corresponds to 1/3 in your correspondence. You
> replied "countable infinity." But "countable infinity" is not a natural
> number.
> Thus 1/3 does not coorespond to anything in your correspondence.
<quote>
> At what point in your list will 1/3 be reached? A rough estimate
> would be acceptable.
When I reach countable infinity.
</quote>
Of course, if I were working in base 3, I could give an exact expression for
1/3. Obviously, I still wouldn't be able to express, say, pi as a finite
string. This does raise the interesting question of what assumptions are
going into Cantor's uncountability proof. The challenge to my generative
algorithm suggests an assumption that 1/3=.333.... That in turn assumes
that it is accepted as meaningful to talk about numbers in terms of an
infinite string of righthand decimal places. By accepting those ideas,
we've already accepted a notion of a completed infinite process. To wit,
that process which generated the string .333..., ad infinitum. One might
attempt to argue that .333..., falls out of Plato's Heaven in one piece
since it can be expressed in closed form. Nonetheless, it seems the very
concept requires a notion of repeated application of a recursive procedure.
Clearly, my algorithm would never have completed the generation of .333....
in a finite number of steps. But neither will Cantor's algorithm enumerate
the rational numbers in a finite number of steps. My algorithm was
performing both the infinite generation of strings such as .333..., and of
all the other real numbers simultaneously (in so much as it is meaning full
to apply the term simultaneous to the collection of steps constituting one
iteration of the outer loop.)
> In other enumeration is a list in which you know that anything in your
> list will eventually appear after going a finite amount along the list.
> You might not know a priori how far along the list you have to go
> before you find an item you are looking for, but you do know that if you
> keep going, you will get to it.
>
> Thus saying you get to 1/3 after countable infinity does not count. For
> it to count, you have to know that 1/3 will appear at some finite place
> along the list. It might be unusually large, but it still has to be
> finite.
Exactly. Doesn't Cantor assume we have already generated that infinite
string to the right of the '.' when he asks us to contemplate the real
numbers between 0 and 1 in decimal point form? (I note that is
not /exactly/ what he does in the paper I provided a link to above.)
>>
>>>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.
>>
>>
>> Or, if I flat-out assert that it is a logical error to assume the
>> completion
>> of an iterative Dedekind mapping, it Cantor's proof becomes an error. Of
>> course, taking pot-shots at infinite iteration, in genera, is a dangerous
>> game. Riemann sums, and many other useful notions become vulnerable.
>
> I'm not familiar with the term "Dedekind mapping."
See Dedekind's definition of an infinite set.
>> I don't believe you fully appreciated what I was getting at. Suppose you
>> have a strip of cloth with buttons along one side and buttonholes along
>> the other.
>>
>> IIIIII
>> 000000
>>
>> Label the buttonholes consecutively using the natural numbers. Label the
>> buttons with the consecutive results of Cantor's first diagonal
>> algorithm. After some finite number of iterations, identify each button
>> that has an integer value assigned to it, and button it to the
>> correspondingly labeled
>> button hole. Notice that the number of unmatched buttons continues to
>> increase per integer button. Each integer button will add on more
>> unmatched button than its predecessor. In the realm of finite sets you
>> will run out of buttons long before you run out of button holes. In
>> infinite sets you never encounter that problem because the game never
>> ends. If you are talking about infinite sets, and using a Dedekind
>> recursive mapping, it is an error to draw a conclusion by contemplating
>> the result of having completed the mapping.
>
> I disagree. No matter how many buttonholes you go along, you will
> always find a button that matches it, even if you have to go for a lot
> longer. And vice versa.
In a finite situation?
>> I don't follow your reasoning here. Why should proof of the consistency
>> of
>> ZF as a result of NBG prove ZF is inconsistent by Gödel's Theorem? The
>> informal formulation of Gödel's result which I remember is that 'no
>> sufficiently powerful system can be shown to be both consistent and
>> complete'.
>>
>
> What it says is that no sufficiently powerful system can be shown to be
> consistent and complete within that system, if that system is
> consistent. For example, it is easy to show Peano's axioms are
> consistent using ZF.
I believe there some commonality between this situation and the observation
that a description of any non-euclidian Riemannian n-space requires an n+1
embedding space (at least locally). That may appear to be amateurish
speculation, but it is based on more than mere fancy. It has to do with
the notion of parameterized parameters.
Also, in C++, at the language level, such things are called templates -
which are generally considered as parameterized types. As I understand
things, Gödel's proof applies to logic of the second order. That is, logic
involving quantified predicates. The similarity between second order logic
and template meta-programming is striking.
--
Nil conscire sibi
Stephen Montgomery-Smith
Cantor's algorithm will not complete counting all the rational numbers
in a finite number of steps. But it will complete counting each
rational number in a finite number of steps (where the finite number
will vary depending on the rational number chosen).
But your algorithm for counting all real numbers will not only not count
all real numbers after finitely many steps, it will also not have
counted the particular real number 1/3 after finitely many steps.
In effect, all you have proven is that there is a countable dense subset
of the real numbers, that is, you can enumerate a set of real numbers
that comes arbitrarily close to every real number.
>
>
>>In other enumeration is a list in which you know that anything in your
>>list will eventually appear after going a finite amount along the list.
>> You might not know a priori how far along the list you have to go
>>before you find an item you are looking for, but you do know that if you
>>keep going, you will get to it.
>>
>>Thus saying you get to 1/3 after countable infinity does not count. For
>>it to count, you have to know that 1/3 will appear at some finite place
>>along the list. It might be unusually large, but it still has to be
>>finite.
>
>
> Exactly. Doesn't Cantor assume we have already generated that infinite
> string to the right of the '.' when he asks us to contemplate the real
> numbers between 0 and 1 in decimal point form? (I note that is
> not /exactly/ what he does in the paper I provided a link to above.)
Cantor assumes (before establishing the contradiction) that there is a
list of real numbers such that each real number (between 0 and 1) will
appear on the list if I look far enough but finitely far along it.
Now each real number also happens to have an infinite decimal
representation. But this doesn't mean that there is an infinite-th
digit, only that for any given natural number n that there is a digit in
the nth place.
>
>
>
>>>>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.
>>>
>>>
>>>Or, if I flat-out assert that it is a logical error to assume the
>>>completion
>>>of an iterative Dedekind mapping, it Cantor's proof becomes an error. Of
>>>course, taking pot-shots at infinite iteration, in genera, is a dangerous
>>>game. Riemann sums, and many other useful notions become vulnerable.
>>
>>I'm not familiar with the term "Dedekind mapping."
>
>
> See Dedekind's definition of an infinite set.
I understand now.
>
>
>>>I don't believe you fully appreciated what I was getting at. Suppose you
>>>have a strip of cloth with buttons along one side and buttonholes along
>>>the other.
>>>
>>>IIIIII
>>>000000
>>>
>>>Label the buttonholes consecutively using the natural numbers. Label the
>>>buttons with the consecutive results of Cantor's first diagonal
>>>algorithm. After some finite number of iterations, identify each button
>>>that has an integer value assigned to it, and button it to the
>>>correspondingly labeled
>>>button hole. Notice that the number of unmatched buttons continues to
>>>increase per integer button. Each integer button will add on more
>>>unmatched button than its predecessor. In the realm of finite sets you
>>>will run out of buttons long before you run out of button holes. In
>>>infinite sets you never encounter that problem because the game never
>>>ends. If you are talking about infinite sets, and using a Dedekind
>>>recursive mapping, it is an error to draw a conclusion by contemplating
>>>the result of having completed the mapping.
>>
>>I disagree. No matter how many buttonholes you go along, you will
>>always find a button that matches it, even if you have to go for a lot
>>longer. And vice versa.
>
>
> In a finite situation?
You give me a finite number of buttons - I can find you a finite number
of buttonholes to match them. It might take me much longer, and I don't
allow you to place any a priori finite limit on how long I am allowed to
look, but I will find the requisite number after a finite amount of time.
And if in response you now give me even more buttons, I can still find
more buttonholes.
>
>
>>>I don't follow your reasoning here. Why should proof of the consistency
>>>of
>>>ZF as a result of NBG prove ZF is inconsistent by Gödel's Theorem? The
>>>informal formulation of Gödel's result which I remember is that 'no
>>>sufficiently powerful system can be shown to be both consistent and
>>>complete'.
>>>
>>
>>What it says is that no sufficiently powerful system can be shown to be
>>consistent and complete within that system, if that system is
>>consistent. For example, it is easy to show Peano's axioms are
>>consistent using ZF.
>
>
> I believe there some commonality between this situation and the observation
> that a description of any non-euclidian Riemannian n-space requires an n+1
> embedding space (at least locally). That may appear to be amateurish
> speculation, but it is based on more than mere fancy. It has to do with
> the notion of parameterized parameters.
I think that if the analogue is interpreted very loosely, that it is fair.
>
> Also, in C++, at the language level, such things are called templates -
> which are generally considered as parameterized types. As I understand
> things, Gödel's proof applies to logic of the second order. That is, logic
> involving quantified predicates. The similarity between second order logic
> and template meta-programming is striking.
Goedel's argument is usually stated for first order logic, although no
doubt it also applies to second order logic.
Gene Ward Smith
Stephen Montgomery-Smith wrote:
> Goedel's argument is usually stated for first order logic, although no
> doubt it also applies to second order logic.
It does. For example, second order arithmetic can be thought of as a
strong version of arithmetic, but equivalently as in effect a weak
version of set theory. Despite the fact that ZFC is a first-order
theory, and NBG a first-order theory which can be finitely axiomized,
they are stronger theories in terms of what can be proven than second
order arithmetic. All of these theories are much stronger than needed
to invoke Goedel incompleteness.
Lasse
Hatto von Aquitanien wrote:
> I have a problem with the idea of assuming that a 1-1 mapping between
> rational numbers and the natural numbers has been established.
This is completely irrelevant to the uncountability of the real
numbers. I sketched a proof that the power set of the natural numbers
is not countable. Do you agree with this proof?
> iterative construction of all
> numbers expressed in decimal notation
Decimal notation is a red herring in these discussions if you ask me.
While it is useful for describing the magnitude of a number, it is not
really very interesting mathematically.
In any case, regarding the countability of the rational numbers:
a) The set of pairs of integers is countably infinite. It is easy to
produce an explicit bijection. Is this something you also do not
understand? Or are you OK with this?
b) The rational numbers can be understood as a subset of the set of
pairs (namely those consisting of two numbers coprime to each other).
So by a), there is a bijective mapping between the set of rational
numbers and a subset of the set of natural numbers.
c) Every subset of the natural numbers is countable.
If you have an objection to either of these points, please state which,
and what your objection is.
Let me make a final remark. You clearly have not studied mathematics
formally. That is not a problem. However, in view of the fact that
thousands of professional mathematicians have studied these matters
before, and have not found a mistake, a little bit of humility would be
befitting. Say 'I don't understand this', rather than 'I don't believe
this'.
Lasse
Hatto von Aquitanien
> Hatto von Aquitanien wrote:
>> Clearly, my algorithm would never have completed the generation of
>> .333....
>> in a finite number of steps. But neither will Cantor's algorithm
>> enumerate
>> the rational numbers in a finite number of steps. My algorithm was
>> performing both the infinite generation of strings such as .333..., and
>> of all the other real numbers simultaneously (in so much as it is meaning
>> full to apply the term simultaneous to the collection of steps
>> constituting one iteration of the outer loop.)
>
> Cantor's algorithm will not complete counting all the rational numbers
> in a finite number of steps. But it will complete counting each
> rational number in a finite number of steps (where the finite number
> will vary depending on the rational number chosen).
>
> But your algorithm for counting all real numbers will not only not count
> all real numbers after finitely many steps, it will also not have
> counted the particular real number 1/3 after finitely many steps.
So in the case of my algorithm, I can always find the serial number for the
generated string, so I could argue that after it completed, I could find
the string .333..., and find its serial number. The problem is that it is
absurd to talk about the process completing. I agree that it is absurd,
but I have to ask why my - intentionally problematic - algorithm is more
problematic than the arguments used in, for example, taking the limit of
Riemann sums as the increments tend to 0. Please don't misunderstand me.
I'm not going to stop integrating functions because I can't convince myself
that it's logically sound. I take it on faith that it is.
> In effect, all you have proven is that there is a countable dense subset
> of the real numbers, that is, you can enumerate a set of real numbers
> that comes arbitrarily close to every real number.
So induction tells us that there is an n-th step such that every string
already generated is serialized, and we can execute the n+1 iteration with
the same result. OTOH, it tells us that every string generated is finite,
and the same is true for n+1.
>> Exactly. Doesn't Cantor assume we have already generated that infinite
>> string to the right of the '.' when he asks us to contemplate the real
>> numbers between 0 and 1 in decimal point form? (I note that is
>> not /exactly/ what he does in the paper I provided a link to above.)
>
> Cantor assumes (before establishing the contradiction) that there is a
> list of real numbers such that each real number (between 0 and 1) will
> appear on the list if I look far enough but finitely far along it.
>
> Now each real number also happens to have an infinite decimal
> representation. But this doesn't mean that there is an infinite-th
> digit, only that for any given natural number n that there is a digit in
> the nth place.
Cantor's cat? It seems as if we are switching between concepts of infinity.
Why can't I say "before you look at the n-th digit, you have to execute my
algorithm at least n times?
>>>I disagree. No matter how many buttonholes you go along, you will
>>>always find a button that matches it, even if you have to go for a lot
>>>longer. And vice versa.
>>
>>
>> In a finite situation?
>
> You give me a finite number of buttons - I can find you a finite number
> of buttonholes to match them. It might take me much longer, and I don't
> allow you to place any a priori finite limit on how long I am allowed to
> look, but I will find the requisite number after a finite amount of time.
>
> And if in response you now give me even more buttons, I can still find
> more buttonholes.
Let me try this a different way since my original example may have been
misleading. Assume two strips of cloth both of the same length. One has n
buttons, the other n button holes. Label the buttons with rational numbers
produced by the first Cantor first diagonal procedure (that's what Hermes
and Markwald call it.) Now fasten a the buttons labeled with integers to
the corresponding button holes. IMPORTANT: leave the other buttons
dangling. If a button which does not have an integer value lies between
two buttons which are already fastened, we consider that button as well as
its nearest integer neighbors to be consumed.
But I have to say, my objection is no longer persuasive to me. I now
remember how I convinced myself, years ago, that Cantor made sense in this
regard. We stop thinking about the numbers being studied as tools to keep
track of the things we are studying. They are simply things being counted,
they are not being used to record the count. For that, we use a different
set of numbers.
That doesn't mean I am convinced that Cantor's proofs are sound and valid.
I still need to give this a lot more thought.
>> Also, in C++, at the language level, such things are called templates -
>> which are generally considered as parameterized types. As I understand
>> things, Gödel's proof applies to logic of the second order. That is,
>> logic involving quantified predicates. The similarity between second
>> order logic and template meta-programming is striking.
>
>
> Goedel's argument is usually stated for first order logic, although no
> doubt it also applies to second order logic.
The impression I have is that in order to completely and consistently
specify any system one needs a system at a higher level of abstraction.
That would suggest a logic of the third order, though I don't know what
that means right now. I should be reading about this rather than posting
about it - the section I am currently working on is about Gödel's proof.
To put this back in terms of C++, there is something people have been
working on called concept checking which is -loosely speaking - a way of
specifying types for type parameters.
--
Nil conscire sibi
albs...@gmx.de
Lasse 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?
>
> No. It is impossible to 'refute' Cantor's proof, since it is correct.
> If you want to convince yourself of this, you should go through the
> proof, and see that indeed each step is justified by the basic axioms
> of set theory.
The fault lies prior Cantor's Diagonalargument.
The problems arise with the declaration of the existence of infinite
sets (Axiom of Infinity).
>
> Of course, nothing stops you from choosing to not believe in some of
> these axioms, and in fact the "existence" of the real numbers
> themselves. However, that is not a "refutation" of Cantor's proof.
There is no choose to believe in axioms since axioms must be consistent
with the used logic. The axiom of infinity is not consistent with our
standard logic. We don't know actual infinity.
Best regards
Albrecht S. Storz
Lasse
> > If you want to convince yourself of this, you should go through the
> > proof, and see that indeed each step is justified by the basic axioms
> > of set theory.
>
> The fault lies prior Cantor's Diagonalargument.
> The problems arise with the declaration of the existence of infinite
> sets (Axiom of Infinity).
No, there is no *fault*, or *problem*. The axiom of infinity is part of
the standard mathematical axioms. Of course, you may choose to
disbelieve the axiom of infinity. Then, however, the question whether
the real numbers are uncountable becomes meaningless, since you won't
even believe in their existence.
> > Of course, nothing stops you from choosing to not believe in some of
> > these axioms, and in fact the "existence" of the real numbers
> > themselves. However, that is not a "refutation" of Cantor's proof.
>
> There is no choose to believe in axioms since axioms must be consistent
> with the used logic. The axiom of infinity is not consistent with our
> standard logic. We don't know actual infinity.
Consistency has a very precise meaning in logic. No inconsistency is
known in the standard axioms of mathematics (say, in ZFC set theory),
and none is believed to exist. Again, it is your *choice* whether you
want to accept the axiom of infinity. But there is no *mistake* in the
proof of the uncountability of the reals, which takes place in the
usual framework of mathematical logic.
Lasse
Hatto von Aquitanien
>
> Hatto von Aquitanien wrote:
>> I have a problem with the idea of assuming that a 1-1 mapping between
>> rational numbers and the natural numbers has been established.
>
> This is completely irrelevant to the uncountability of the real
> numbers.
I now appreciate the difference.
> I sketched a proof that the power set of the natural numbers
> is not countable. Do you agree with this proof?
Unless you post under a different nym, I have not seen this.
>> iterative construction of all
>> numbers expressed in decimal notation
>
> Decimal notation is a red herring in these discussions if you ask me.
> While it is useful for describing the magnitude of a number, it is not
> really very interesting mathematically.
Since the form of Cantor's second diagonal procedure that I was responding
to used decimal notation it seemed appropriate to used the same formalism.
I don't believe it is a red herring. That would imply it is an irrelevant
distraction. Insisting that Cantor's proof be formulated in decimal
notation would be a red herring. Calling the use of decimal notation a red
herring could justly be considered a red herring.
> In any case, regarding the countability of the rational numbers:
>
> a) The set of pairs of integers is countably infinite. It is easy to
> produce an explicit bijection. Is this something you also do not
> understand? Or are you OK with this?
In simple terms I can read the words, and know what they mean. I can follow
the induction which shows that for every iteration the mapping remains
bijective.
> b) The rational numbers can be understood as a subset of the set of
> pairs (namely those consisting of two numbers coprime to each other).
> So by a), there is a bijective mapping between the set of rational
> numbers and a subset of the set of natural numbers.
Assuming that it is meaningful to talk about mapping infinite sets into or
onto infinite sets.
> c) Every subset of the natural numbers is countable.
>
> If you have an objection to either of these points, please state which,
> and what your objection is.
In the most primitive sense of "countable", this is actually not true. No
human, and no computer can count an infinite number of things.
> Let me make a final remark. You clearly have not studied mathematics
> formally.
My degree is in computer science.
> That is not a problem. However, in view of the fact that
> thousands of professional mathematicians have studied these matters
> before, and have not found a mistake, a little bit of humility would be
> befitting.
It would appear that not all are in agreement on this point. It is
certainly conceivable that an academic discipline has a cult of intolerance
which automatically rejects any challenge to its orthodoxy.
> Say 'I don't understand this', rather than 'I don't believe this'.
What I actually said is that I have never been persuaded of this. Perhaps a
bit of accuracy on your part is in order?
--
Nil conscire sibi
Dave Seaman
> Lasse wrote:
>> Decimal notation is a red herring in these discussions if you ask me.
>> While it is useful for describing the magnitude of a number, it is not
>> really very interesting mathematically.
> Since the form of Cantor's second diagonal procedure that I was responding
> to used decimal notation it seemed appropriate to used the same formalism.
> I don't believe it is a red herring. That would imply it is an irrelevant
> distraction. Insisting that Cantor's proof be formulated in decimal
> notation would be a red herring. Calling the use of decimal notation a red
> herring could justly be considered a red herring.
It doesn't matter whether the base is decimal or not. The point is that
using any base at all, i.e., representing real numbers as strings of
digits, is the red herring. Cantor did not use such a representation in
his first proof of the uncountability of the reals, and the proof of the
uncountability of P(N) likewise does not depend on decimal (or any other
base) notation.
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
Dave L. Renfro
> Of course, if I were working in base 3, I could give an
> exact expression for 1/3. Obviously, I still wouldn't be
> able to express, say, pi as a finite string. This does
> raise the interesting question of what assumptions are
> going into Cantor's uncountability proof. The challenge
> to my generative algorithm suggests an assumption that
> 1/3=.333.... That in turn assumes that it is accepted
> as meaningful to talk about numbers in terms of an infinite
> string of righthand decimal places. By accepting those
> ideas, we've already accepted a notion of a completed
> infinite process. To wit, that process which generated
> the string .333..., ad infinitum.
Infinite decimals as "completed infinities" had an
important role in the 1870-1890 (roughly) development
of an arithmetic foundation for the real numbers.
[The term "arithmetic" in this sense means not based
on geometry, but based in a purely logical/mathematical
way on the integers.] Cantor was also one of the key
participants in this process. However, if we're only
talking about the positive rational numbers, this is
not a concern.
It is possible to describe rational numbers in a
"finite way" so that we don't have to worry about
completed infinities. Specifically, each positive
rational number can be expressed as the ratio of
two positive integers. To wit, as 'm/n'. Note that
it only takes a finite number of characters to
write this -- the finitely many digits needed
to write 'm', the fraction line '/', and the
finitely many characters needed to write 'n'.
Using these "finite descriptions", we can make
a list of all the rational numbers between 0 and 1
very easily. First, arrange all such numbers into
rows with a common denominator:
0/1 1/1
0/2 1/2 2/2
0/3 1/3 2/3 3/3
0/4 1/4 2/4 3/4 4/4
0/5 1/5 2/5 3/5 4/5 5/5
.
.
.
Now construct the list by listing the numbers going
across the 1'st row, followed by the numbers going
across the 2'nd row, followed by the numbers going
across the 3'rd row, followed by the numbers going
across the 4'th row, etc.
If anything, there's an "overkill" going on, since
we're actually listing each rational number between
0 and 1 infinitely many times (e.g. 1/3, 2/6, 3/9, ...).
If you want a true 1-1 correspondence, then skip over
any fractions coming your way that aren't in reduced
form (and consider 0/1 to be the reduced form of 0).
Some more work is needed to get a list of all positive
rational numbers, but I think many of your concerns
apply to just being able to get a list of those
between 0 and 1, and the above method does this.
Dave L. Renfro
Hatto von Aquitanien
> Hatto von Aquitanien wrote (in part):
> Some more work is needed to get a list of all positive
> rational numbers, but I think many of your concerns
> apply to just being able to get a list of those
> between 0 and 1, and the above method does this.
I have not questioned the methodology shown here as a means of visiting each
rational number sequentially:
http://www.homeschoolmath.net/teaching/rational-numbers-countable.php
My discussion of numbers between 0 and 1 is a result of the presentation
provided by Hermes and Markwald. One point which I had misunderstood was
the correlation between the first and second diagonal methods. Or better,
I had assumed a correlation when none was intended. The presentation is
very - I mean to say *extremely* - condensed. The constructive feedback I
have received in this discussion helped me realize that error.
My misgivings are more along the lines of what it all really means, and what
assumptions are actually made at the outset. Hopefully it will be a bit
clearer when I read the chapter on constructing the real numbers. It has
an appendix devoted to ordinal numbers.
--
Nil conscire sibi
Hatto von Aquitanien
> On Mon, 10 Jul 2006 08:34:57 -0400, Hatto von Aquitanien wrote:
>> Lasse wrote:
>
>>> Decimal notation is a red herring in these discussions if you ask me.
>>> While it is useful for describing the magnitude of a number, it is not
>>> really very interesting mathematically.
>
>> Since the form of Cantor's second diagonal procedure that I was
>> responding to used decimal notation it seemed appropriate to used the
>> same formalism.
>> I don't believe it is a red herring. That would imply it is an
>> irrelevant
>> distraction. Insisting that Cantor's proof be formulated in decimal
>> notation would be a red herring. Calling the use of decimal notation a
>> red herring could justly be considered a red herring.
>
> It doesn't matter whether the base is decimal or not. The point is that
> using any base at all, i.e., representing real numbers as strings of
> digits, is the red herring. Cantor did not use such a representation in
> his first proof of the uncountability of the reals, and the proof of the
> uncountability of P(N) likewise does not depend on decimal (or any other
> base) notation.
Let m = 0 and w = 1.
http://uk.geocities.com/frege%40btinternet.com/cantor/diagarg.htm
--
Nil conscire sibi
Dave Seaman
That's Cantor's second proof. I was talking about the first one.
See <http://en.wikipedia.org/wiki/Cantor's_first_uncountability_proof>.
Dik T. Winter
...
You fall in the same trap that Mueckenheim did fall in in another thread.
The sets E are *not* binary numbers. For one thing, the sets:
E1 = (m, w, w, w, w, ...)
is different from
E2 = (w, m, m, m, m, ...)
but when you consider them as binary numbers they are identical.
So those sets are *not* binary numbers.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Stephen Montgomery-Smith
Despite many people calling you a crank, I really do believe that you
are on the cusp of "getting it." The misconceptions you have are ones I
used to have before I studied pure math more deeply, so I do feel I
understand where you are coming from. One recommendation I have - you
might like to take a rest from arguing these issues on sci.math for a
while. Give yourself time to reflect on the issues. You come across to
me as a bright and honest person, and so I think that you will really
get it.
Best Stephen
Lasse
> > is not countable. Do you agree with this proof?
>
> Unless you post under a different nym, I have not seen this.
It was posted, under the same name, in this same thread. If you use
e.g. Google Groups, you can find the posting.
> > Decimal notation is a red herring in these discussions if you ask me.
> > While it is useful for describing the magnitude of a number, it is not
> > really very interesting mathematically.
>
> Since the form of Cantor's second diagonal procedure that I was responding
> to used decimal notation it seemed appropriate to used the same formalism.
> I don't believe it is a red herring. That would imply it is an irrelevant
> distraction. Insisting that Cantor's proof be formulated in decimal
> notation would be a red herring. Calling the use of decimal notation a red
> herring could justly be considered a red herring.
Decimal notation (or, in fact, the use of any other base) includes
several subtleties, such as the nuisance that different sequences of
natural numbers can represent the same real number (e.g., 0.19999... =
0.20000...). It is conceptually much cleaner to decide first that the
power set of N is not countable, and then see that there are at least
as many real numbers as subsets of N.
> > In any case, regarding the countability of the rational numbers:
> >
> > a) The set of pairs of integers is countably infinite. It is easy to
> > produce an explicit bijection. Is this something you also do not
> > understand? Or are you OK with this?
>
> In simple terms I can read the words, and know what they mean. I can follow
> the induction which shows that for every iteration the mapping remains
> bijective.
I am not quite sure what you mean by this. The bijection can be
visualized nicely. For example, to find a bijection betwen N and N^2,
draw N^2 as a grid in the plane, and think of counting the numbers
along the diagonals:
(0,0) --- (1,0) , (0,1) --- (2,0) , (1,1) , (0,2) --- ...
Similarly, you can draw Z^2 as a grid in the plane, and move outwards
in a spiral-like pattern, etc ...
> > b) The rational numbers can be understood as a subset of the set of
> > pairs (namely those consisting of two numbers coprime to each other).
> > So by a), there is a bijective mapping between the set of rational
> > numbers and a subset of the set of natural numbers.
>
> Assuming that it is meaningful to talk about mapping infinite sets into or
> onto infinite sets.
Of course it is meaningful in the standard context of set theory.
> > c) Every subset of the natural numbers is countable.
> >
> > If you have an objection to either of these points, please state which,
> > and what your objection is.
>
> In the most primitive sense of "countable", this is actually not true. No
> human, and no computer can count an infinite number of things.
"Countable" is a mathematical term, with a precise definition, as I am
sure you are well-aware. I.e., a set M is countable if there is an
injective mapping from M into the natural numbers.
> > Let me make a final remark. You clearly have not studied mathematics
> > formally.
>
> My degree is in computer science.
In my time, computer science studies included a good deal of formal
mathematics, including a sound background in logic, which should be
more than enough to deal with these questions. Perhaps this is no
longer the case.
>
> It would appear that not all are in agreement on this point. It is
> certainly conceivable that an academic discipline has a cult of intolerance
> which automatically rejects any challenge to its orthodoxy.
Oh, it's certainly conceivable. However, in a field like mathematics,
and at this basic level (which thousands of students encounter every
year), you will have to agree that this is highly unlikely (to make an
understatement). Generally, mathematics is fairly robust with respect
to not producing results that are wrong: proofs can, and are, generally
checked quite thoroughly, particularly since people will want to apply
the same methods. As subfields get more specialized, there is more
danger that errors might not be spotted (particularly when the proofs
are not well-written), but we are not discussing a result at that
level.
Suggesting that Cantor's diagonal proof is wrong is like saying that
there has never been a painting called the Mona Lisa, and that there is
only an empty frame hanging in the Louvre. Wouldn't the millions of
visitors have noticed?
>
> > Say 'I don't understand this', rather than 'I don't believe this'.
>
> What I actually said is that I have never been persuaded of this. Perhaps a
> bit of accuracy on your part is in order?
You said that you want to "refute" Cantor's proof. You did not say that
you want to "understand" Cantor's proof. If that is not (or no longer)
the case, all the better --- but in my sporadic visits to this board, I
have encountered too many people with this kind of attitude. If you are
not one of them, good for you.
Virgil
albs...@gmx.de wrote:
> Lasse 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?
> >
> > No. It is impossible to 'refute' Cantor's proof, since it is correct.
> > If you want to convince yourself of this, you should go through the
> > proof, and see that indeed each step is justified by the basic axioms
> > of set theory.
>
> The fault lies prior Cantor's Diagonalargument.
The fault in "albstorz" understanding "lies prior Cantor's
Diagonalargument".
> The problems arise with the declaration of the existence of infinite
> sets (Axiom of Infinity).
That causes no problems at all for most people, so that problem lies in
the interface between that axiom and "albstorz", and not in the axiom
itself.
>
> >
> > Of course, nothing stops you from choosing to not believe in some of
> > these axioms, and in fact the "existence" of the real numbers
> > themselves. However, that is not a "refutation" of Cantor's proof.
>
> There is no choose to believe in axioms since axioms must be consistent
> with the used logic.
There is ample evidence that people chose believe in all sorts of
inconsistent things. So why would that be a drawback?
And in any case "albstorz" has shown no inconsistency withoin, say ,
ZFC of NBG. only incompatibility between them and what "albstorz"
believes. And "albstorz" cannot demonstrate that it is not "albstorz"'s
beliefs that are inconsistent with logic.
> The axiom of infinity is not consistent with our
> standard logic.
That the axiom of infinity is incompatible with "albstorz"'s logic, is
"albstorz"'s problem.
> We don't know actual infinity.
Ignorance is no excuse.
Nathan
> I have a problem with the idea of assuming that a 1-1 mapping between
> rational numbers and the natural numbers has been established. That is, the
> step between the first diagonal construction, and the application of the
> second diagonal method. It assumes that it is meaningful to contemplate
> the result of executing an infinite number of constructive steps. Notice
> that when I posted my algorithm for the iterative construction of all
> numbers expressed in decimal notation I was asked when I would get to 1/3.
> My response was, 'when I reach countable infinity'. That was rejected as
> meaningless. Nonetheless, that seems to be exactly what we have to assume
> when we apply Cantor's second diagonal method.
You can't have it both ways. The statement "the set of real numbers is
uncountable" means there exists no surjection from the set of integers
to the set of reals. If you don't accept the idea that a function can
be defined on an infinite domain, then you must believe that the reals
are uncountable.
David R Tribble
>> It also maps bijectively to "completed infinity". There seems to be a
>> problem here. For every rational number we previously claimed there was a
>> corresponding integer.
>
Matt wrote:
> That's right. There are various systematic ways to list all the
> rational numbers. I recall there's a quite simple iterative way to do
> it so that each rational number appears exactly once, in its simplest
> form, though I can't immediately find a reference. Thus the rational
> numbers can be put into a one-to-one correspondence with the natural
> numbers, and are countable (unlike the irrationals).
A simple mapping, which I [re]discovered a few months ago:
S(0) = 0;
S(1) = 1;
S(2n) = S(n)+1, for n > 0;
S(2n+1) = 1/S(2n).
The inverse mapping is:
iS(0) = 0;
iS(1) = iS(p/p) = 1, for p > 0;
iS(p) = iS(p/1) = 2p-1, for p > 0;
iS(p/q) = 2iS(p/q -1), if p > q;
iS(p/q) = iS(q/p) + 1, if p < q.
This maps every natural to every positive rational (in simplest form),
and vice versa.
Enhancing this to include the negative rationals and/or the negative
integers is fairly trivial.
(Google for "Farey sequence".)
David R Tribble
>> Of course, nothing stops you from choosing to not believe in some of
>> these axioms, and in fact the "existence" of the real numbers
>> themselves. However, that is not a "refutation" of Cantor's proof.
>
Albrech Storz wrote:
> There is no choose to believe in axioms since axioms must be consistent
> with the used logic. The axiom of infinity is not consistent with our
> standard logic. We don't know actual infinity.
I have set S such that:
1. If n is in S, then n+1 is in S;
2. 0 is in S.
Why is S not an "actually" infinite set?
David R Tribble
> I have a problem with the idea of assuming that a 1-1 mapping between
> rational numbers and the natural numbers has been established. That is, the
> step between the first diagonal construction, and the application of the
> second diagonal method. It assumes that it is meaningful to contemplate
> the result of executing an infinite number of constructive steps. Notice
> that when I posted my algorithm for the iterative construction of all
> numbers expressed in decimal notation I was asked when I would get to 1/3.
> My response was, 'when I reach countable infinity'. That was rejected as
> meaningless. Nonetheless, that seems to be exactly what we have to assume
> when we apply Cantor's second diagonal method.
>
> So I ask, if Cantor can present his constructive method and then have us
> contemplate the result of its completion, why can I not do the same?
For the simple reason that Cantor's proof assumes that the starting
list is countable, which is the entire crux of the proof. If you
assume you have a countable list of all reals, it can be proved that at
least one real is missing from the list, hence there is no exhaustive
countable list of all the reals.
It has nothing to do with how "long" it takes to make the list or to
construct the diagonal value. As soon as you postulate the
existence of the list, the diagonal value exists as well, and that
value is always missing from the list.
Ross A. Finlayson
Basically, you're referring to three different, though related, things:
1) the antidiagonal result of coded decimal real numbers or coded
powerset (N <=/=> R[0,1], N<=/=> R_2[0,1])
2) the nested intervals result or "Cantor's first proof", or
Cantor/Megill style
3) the powerset result (Exists S s.t. for n E N, if n E f(n) then not n
E S else n E S)
There is no universe in ZF, the theory in which these results are
proferred, so ZF is inconsistent. That implies you can prove anything
in ZF, like how you can apply nested intervals to the rationals, but
different.
If you look at the antidiagonal result, if you consider a list of real
numbers that starts at zero and then in some fantastic way, for example
base one or base infinity, proceeds in the natural order of the real
numbers, then the antidiagonal is off the list, it's simply the end of
the list, where there are various absurdities about there being an end
to an infinite list.
With nested intervals, and there are results in real analysis that
there is least positive/non-negative real numbers, in going about
well-ordering the real numbers, then you get to degenerate or
"indistinguishable" intervals, and where there were in well-ordering
the reals, then there are. Basically the argument goes that the only
way to biject the naturals to the unit interval of reals is by
well-ordering them, and the only way to well-order them is to avoid the
very same conclusions of "Cantor's first" or here called
"Cantor/Megill" as applied to a well-ordering instead of a bijection to
the naturals.
With the powerset result, basically there is vacuity in the successor
operation as f, with ordinals. So, when f(x)=x+1, S = NULL.
Re-Vitali-ize measure theory. Impulse functions: real. There are a
variety of small errors in Hodges' paper, of Hodges.
Quantify over sets: there is no universe in ZF. Would you like to
read some considerations about Cantor's theory? Read Cantor. He says
there is a universe, so ZF is anti-Cantorian, in the sense that for not
being naive it's not particularly sophisticated. What is M the model
in model theory? There is no greatest ordinal in ZF. There are no
non-set classes in pure set theory and no non-theory models in pure
theory."
There's only one theory with no axioms.
Ross
Jonathan Hoyle
Correct. It is the one with no theorems.
Tony Orlow
> Hatto von Aquitanien wrote:
>> I have a problem with the idea of assuming that a 1-1 mapping between
>> rational numbers and the natural numbers has been established. That is, the
>> step between the first diagonal construction, and the application of the
>> second diagonal method. It assumes that it is meaningful to contemplate
>> the result of executing an infinite number of constructive steps. Notice
>> that when I posted my algorithm for the iterative construction of all
>> numbers expressed in decimal notation I was asked when I would get to 1/3.
>> My response was, 'when I reach countable infinity'. That was rejected as
>> meaningless. Nonetheless, that seems to be exactly what we have to assume
>> when we apply Cantor's second diagonal method.
>>
>> So I ask, if Cantor can present his constructive method and then have us
>> contemplate the result of its completion, why can I not do the same?
>
> For the simple reason that Cantor's proof assumes that the starting
> list is countable, which is the entire crux of the proof. If you
> assume you have a countable list of all reals, it can be proved that at
> least one real is missing from the list, hence there is no exhaustive
> countable list of all the reals.
I'm sorry, David, but it assumes more than that. It assumes that such a
list is diagonally traversable, a fact which is contradicted directly by
N=S^L for S>1. This proof is not possible in unary, that is, for S=1,
because it becomes immediately obvious that the antidiagonal is either
the last finite, which doesn't "exist", or the very next element in the
list. Using binary, decimal, or any base greater than 1, we indeed find
that the width of any list becomes exponentially smaller than its
length, as n increases.
If, and this is a big "if", we assume that there is this particular
number of bits, one for each natural number, and call it omega or
aleph_0, then it stands to reason that strings of such a length would
have an exponentially greater number than this length has positions.
This is the conrerstone of distinguishing infinities. But, it's only the
beginning.
>
> It has nothing to do with how "long" it takes to make the list or to
> construct the diagonal value. As soon as you postulate the
> existence of the list, the diagonal value exists as well, and that
> value is always missing from the list.
>
That's because the diagonal doesn't fully traverse the list. In binary,
when you traverse n columns and n rows diagonally, there are fully 2^n
rows possible with those n bits, so of course you've missed the vast
majority at any given point for n>0. In decimal it gets worse. So what?
Yes, the power set, or digital representation (same thing, ultimately)
is larger than the set. It can also be bijectable. :)
Tony Orlow
> Hi,
>
> Basically, you're referring to three different, though related, things:
>
> 1) the antidiagonal result of coded decimal real numbers or coded
> powerset (N <=/=> R[0,1], N<=/=> R_2[0,1])
Decimal, binary, power set, digital. Yes. I have come recently to the
realization that the probabilistic generalization of logic, either in a
finite number of discrete values or continuously, leads to the
generalization of power set, such that it subsumes the digital number
system.
> 2) the nested intervals result or "Cantor's first proof", or
> Cantor/Megill style
Yes, this is a different style of the proof. I actually don't have any
fundamental problem with this proof, given the definition of "countable"
as every being at a finite position in the set. There is no doubt in my
mind that this proof is essentially sound in demonstrating that, to
represent every real number in any finite interval requires an infinite
number of bits. That is, there will always be elements that require more
than any finite number of steps to specify, symbolically, in language.
Take, for example. 1/3 in decimal. :)
> 3) the powerset result (Exists S s.t. for n E N, if n E f(n) then not n
> E S else n E S)
Again, power set, equivalent to the digital representation. This is why
I talk about N=S^L separately from IFR (though I often enough mention
them together, as a reminder ;). One deals with the discrete and
combinatorial, and the other with spaces and measure boiled down to units.
>
> There is no universe in ZF, the theory in which these results are
> proferred, so ZF is inconsistent. That implies you can prove anything
> in ZF, like how you can apply nested intervals to the rationals, but
> different.
Hi Ross - How are you? It's nice to see you in the Pit. :)
Now, what IS the Universe? Do you have a concept where to start? I know
you do. :)
Is it the closure of all possible operations?
That pot's starting to smell good, but I think it needs some bay leaf
and a clove, or a head, more garlic...
>
> If you look at the antidiagonal result, if you consider a list of real
> numbers that starts at zero and then in some fantastic way, for example
> base one or base infinity, proceeds in the natural order of the real
> numbers, then the antidiagonal is off the list, it's simply the end of
> the list, where there are various absurdities about there being an end
> to an infinite list.
Absolutely. In those two cases, the argument evaporates, Did I say that
to David? Infinite base, as you must imagine, dear Ross, is a little
hard for people to envision. If you're going to mention it, it might
help to refer to the T-riffics, and the fact that between each pair of
limit points is essentially a base-oo digit. But, I suppose it would
help if I wrote up a good page on that for you to cite, eh? It's on my
list. :)
>
> With nested intervals, and there are results in real analysis that
> there is least positive/non-negative real numbers, in going about
> well-ordering the real numbers, then you get to degenerate or
> "indistinguishable" intervals, and where there were in well-ordering
> the reals, then there are. Basically the argument goes that the only
> way to biject the naturals to the unit interval of reals is by
> well-ordering them, and the only way to well-order them is to avoid the
> very same conclusions of "Cantor's first" or here called
> "Cantor/Megill" as applied to a well-ordering instead of a bijection to
> the naturals.
Since any finite number of subdivisions of the unit interval will result
in finite intervals which can be further subdivided, there must be an
infinite number, greater than any finite number, of intermediate points
with which to subdivide nested intervals on the real line.
If we assign some infinite unit X to this infinite number of reals in
the unit interval, indeed, we have a sequence of values, n/X for n e *N,
infinitesimally distinguished, and one of at least two sequential
orderings of the reals. Unfortunately, despite the fact that the Axiom
of Choice (the C in ZFC) affords that there exist uncountable
well-ordered sets, it appears, after some perusal, to be impossible to
define one, except on the set of ordinals. The reason for this is
predecessor discontinuity, or the lack of "infinite descending chains".
>
> With the powerset result, basically there is vacuity in the successor
> operation as f, with ordinals. So, when f(x)=x+1, S = NULL.
Rhat?
>
> Re-Vitali-ize measure theory. Impulse functions: real. There are a
> variety of small errors in Hodges' paper, of Hodges.
I'd have to look that up. Did you turn me on to Chu spaces? That seemed
like a ripe generalization. I'll have to digest it a bit to see how it
goes down. :)
>
> Quantify over sets: there is no universe in ZF. Would you like to
> read some considerations about Cantor's theory? Read Cantor. He says
> there is a universe, so ZF is anti-Cantorian, in the sense that for not
> being naive it's not particularly sophisticated. What is M the model
> in model theory? There is no greatest ordinal in ZF. There are no
> non-set classes in pure set theory and no non-theory models in pure
> theory."
>
> There's only one theory with no axioms.
Does it start with a statement of the dimensions of the space? :)
>
> Ross
>
Happy Summer!
Tony
webs...@gmail.com
> I'm interested to know what attempts have been made to refute Cantor's proof
> that the real numbers are not denumerable?
None of any relevant consequence.
> [...] Quite honestly, I find the second diagonal method unconvincing.
That's nice. I find the fact that G. W. Bush is my president
unconvincing too. But apparently, this is just a dellusion on my part,
and he actually is the president. One day I may be able to face this
reality, but for now, I am just in denial. What kind of medication do
you take for your dellusion?
> [...] 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?
You might like to pursue this with Alex Jones or Michael Rupert. These
are renowned experts in the "research community" that are as likely to
help you as anyone.
--
Paul Hsieh
http://www.pobox.com/~qed/
http://bstring.sf.net/
Tony Orlow
This is very interesting, David. I'm going to have to digest it for a
bit. Every TOE has to take care of everything, right? Thanx. :)
Tony
Hatto von Aquitanien
> You can't have it both ways. The statement "the set of real numbers is
> uncountable" means there exists no surjection from the set of integers
> to the set of reals. If you don't accept the idea that a function can
> be defined on an infinite domain, then you must believe that the reals
> are uncountable.
One question I have with all of this is whether that is a tacit assumption
Cantor made and relied upon for his proof. It seems, if nothing else, he
had a concept of a 2 dimensional infinite grid of tokens. So there appears
to at least be ci X ci where ci means countable infinity.
--
Nil conscire sibi
Virgil
Tony Orlow <ae...@cornell.edu> wrote:
> David R Tribble wrote:
> > Hatto von Aquitanien wrote:
> >> I have a problem with the idea of assuming that a 1-1 mapping between
> >> rational numbers and the natural numbers has been established. That is, the
> >> step between the first diagonal construction, and the application of the
> >> second diagonal method. It assumes that it is meaningful to contemplate
> >> the result of executing an infinite number of constructive steps. Notice
> >> that when I posted my algorithm for the iterative construction of all
> >> numbers expressed in decimal notation I was asked when I would get to 1/3.
> >> My response was, 'when I reach countable infinity'. That was rejected as
> >> meaningless. Nonetheless, that seems to be exactly what we have to assume
> >> when we apply Cantor's second diagonal method.
> >>
> >> So I ask, if Cantor can present his constructive method and then have us
> >> contemplate the result of its completion, why can I not do the same?
> >
> > For the simple reason that Cantor's proof assumes that the starting
> > list is countable, which is the entire crux of the proof. If you
> > assume you have a countable list of all reals, it can be proved that at
> > least one real is missing from the list, hence there is no exhaustive
> > countable list of all the reals.
>
> I'm sorry, David, but it assumes more than that.
TO may assume more than that but Cantor does not.
> >
> > It has nothing to do with how "long" it takes to make the list or to
> > construct the diagonal value. As soon as you postulate the
> > existence of the list, the diagonal value exists as well, and that
> > value is always missing from the list.
> >
>
> That's because the diagonal doesn't fully traverse the list.
Which members of that list does it ski[p over,TO?
The members of the list are in bijective correspondence with the members
of N and for each n in N, the diagaonal is not equal to the real
corresponding to that n.
So which reals in the list did we miss, TO?
Hatto von Aquitanien
> Hatto von Aquitanien wrote:
>> So I ask, if Cantor can present his constructive method and then have us
>> contemplate the result of its completion, why can I not do the same?
>
> For the simple reason that Cantor's proof assumes that the starting
> list is countable, which is the entire crux of the proof.
I thought he assumed it was _counted_ .
> If you
> assume you have a countable list of all reals, it can be proved that at
> least one real is missing from the list, hence there is no exhaustive
> countable list of all the reals.
Originally my (incorrect) understanding was that Cantor first put all the
rational numbers into a counted list, and then proved he hadn't counted
everything. I now realize that was incorrect. However, /suppose/ we /did/
claim that we have a counted list of rational numbers (1-1 correspondence
with N+), but don't explain how we came by it. Would Cantor's diagonal
insertion method not indicate there are always more rational numbers?
This question may in fact be the origin of my previous confusion. I may
have entertained this same question years ago, and never followed up on it.
> It has nothing to do with how "long" it takes to make the list or to
> construct the diagonal value. As soon as you postulate the
> existence of the list, the diagonal value exists as well, and that
> value is always missing from the list.
I trust this has been suggested before, but suppose we are talking about a
two-valued underlying domain, and use Cantor's diagonal method to enumerate
the sequences.
--
Nil conscire sibi
Hatto von Aquitanien
Basically, you've given an algorithm for generating the members of S and
turned it into a predicate. For the sake of keeping track of ideas which
underly the conclusion, should we not consider this concept
of /doing/ /something/ /repeatedly/ as one of the premises?
--
Nil conscire sibi
imagin...@despammed.com
> David R Tribble wrote:
> > Hatto von Aquitanien wrote:
> >> I have a problem with the idea of assuming that a 1-1 mapping between
> >> rational numbers and the natural numbers has been established. That is, the
> >> step between the first diagonal construction, and the application of the
> >> second diagonal method. It assumes that it is meaningful to contemplate
> >> the result of executing an infinite number of constructive steps. Notice
> >> that when I posted my algorithm for the iterative construction of all
> >> numbers expressed in decimal notation I was asked when I would get to 1/3.
> >> My response was, 'when I reach countable infinity'. That was rejected as
> >> meaningless. Nonetheless, that seems to be exactly what we have to assume
> >> when we apply Cantor's second diagonal method.
> >>
> >> So I ask, if Cantor can present his constructive method and then have us
> >> contemplate the result of its completion, why can I not do the same?
> >
> > For the simple reason that Cantor's proof assumes that the starting
> > list is countable, which is the entire crux of the proof. If you
> > assume you have a countable list of all reals, it can be proved that at
> > least one real is missing from the list, hence there is no exhaustive
> > countable list of all the reals.
>
> I'm sorry, David, but it assumes more than that. It assumes that such a
> list is diagonally traversable, a fact which is contradicted directly by
> N=S^L for S>1.
Ah right. But much, much more. The traditional view assumes there is
only one set of reals, whereas it was revealed to me recently that in
fact they come in three types, which for reasons not yet revealed are
known as vanilla, strawberry, and chocolate.
> This proof is not possible in unary, that is, for S=1,
Oh, right. Yes, go on. Tell us more...
> because it becomes immediately obvious that the antidiagonal is either
> the last finite, which doesn't "exist", or the very next element in the
> list.
But if the last finite, which doesn't exist is vanilla, then it can't
be vanilla, can it. And the very next element in the list - I mean, I
have got this right haven't I, this is the very next element after the
last element which doesn't exist. Hum, I'm sure this would be
strawberry.
So the answer is clear: a chocolate sundae. Proved to exist at the end
of the rainbow.
> Using binary, decimal, or any base greater than 1, we indeed find
> that the width of any list becomes exponentially smaller than its
> length, as n increases.
Ah, yes, and tastier too. But as we near the end of the unending
sequence of exponentially different widths and lengths, uh what
happens? The end doesn't exist - the bottom doesn't exist, and the
right edge doesn't exist. That doesn't leave much, does it? Would it be
that somewhere after the end of the unending sequence, chocolate,
vanilla and strawberry all change places, and the result is a sort of
caramel taste. Sort of burning sweetness?
>
> If, and this is a big "if", we assume that there is this particular
> number of bits, one for each natural number, and call it omega or
> aleph_0,
No, don't call this "particular number of bits" omega or aleph_0, since
those are mathematical terms wildly beyond your understanding. Call it
Enrico, Neapolitan vendor since the pisctachio bit the dust in
1950-whatever.
then it stands to reason that strings of such a length would
> have an exponentially greater number than this length has positions.
> This is the conrerstone of distinguishing infinities. But, it's only the
> beginning.
Right. Off you go then. I'm going to have my dinner.
>
> >
> > It has nothing to do with how "long" it takes to make the list or to
> > construct the diagonal value. As soon as you postulate the
> > existence of the list, the diagonal value exists as well, and that
> > value is always missing from the list.
> >
>
> That's because the diagonal doesn't fully traverse the list.
"Traverse"? You mean that something that merely goes on for ever in the
direction of something with no end, may fail to be within the
appropriate fraction of the end that doesn't exist? How would you
calculate such things?
Nuff babble for now...
Brian Chandler
http://imaginatorium.org
Dave Seaman
> Originally my (incorrect) understanding was that Cantor first put all the
> rational numbers into a counted list, and then proved he hadn't counted
> everything. I now realize that was incorrect. However, /suppose/ we /did/
> claim that we have a counted list of rational numbers (1-1 correspondence
> with N+), but don't explain how we came by it. Would Cantor's diagonal
> insertion method not indicate there are always more rational numbers?
No, not at all. The if all the rationals are in the list, then the diagonal
method must necessarily produce an irrational number.
And we don't need to "suppose" we have a counted list of rational numbers.
It's been done. See, for example,
<http://www.lacim.uqam.ca:16080/~plouffe/OEIS/citations/recounting.pdf>.
> This question may in fact be the origin of my previous confusion. I may
> have entertained this same question years ago, and never followed up on it.
>> It has nothing to do with how "long" it takes to make the list or to
>> construct the diagonal value. As soon as you postulate the
>> existence of the list, the diagonal value exists as well, and that
>> value is always missing from the list.
> I trust this has been suggested before, but suppose we are talking about a
> two-valued underlying domain, and use Cantor's diagonal method to enumerate
> the sequences.
A two-valued underlying domain? Isn't that basically happens with the Cantor
proof that |X| < |P(X)|? A subset of X is identified by its characteristic
function, which is a mapping f: X -> {0,1}.
<http://www.mathacademy.com/pr/prime/articles/cantor_theorem/>