A Correction in Set Theory
Dave L. Renfro
the topic of a thread I began on July 3, "MONOTONE
SET FUNCTIONS, FIXED POINTS, AND CECH CLOSURE FUNCTIONS",
is now available on-line: Preston C. Hammer's "General
topology, symmetry, and convexity", Transactions of the
Wisconsin Academy of Sciences, Arts and Letters 44 (1956),
221-255.
This got me thinking ... isn't this the journal that
published William Dilworth's crankish math paper, "A
correction in set theory" (MR 58 #16089)? Yep, it is.
For those who don't know what I'm talking about, this
is the crank paper mentioned in Underwood Dudley's 1992
book "Mathematical Cranks" that disputes Cantor's proof
of the uncountability of the reals and which managed
to get published. Dudley's remarks caused him to be
subsequently sued by Dilworth. For Usenet posts about
this, see <http://tinyurl.com/cffrn>. 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.
Circuit Opinion for Dilworth vs. Dudley:
http://www.law.emory.edu/7circuit/jan96/95-2282.html
On-line copy of William Dilworth's "A correction
in set theory":
http://digicoll.library.wisc.edu/cgi-bin/WI/WI-idx?type=article&byte=10235322
An antidote to Dilworth's paper: Wilfrid Hodges,
"An editor recalls some hopeless papers", Bulletin
of Symbolic Logic 4 (1998), 1-16:
http://www.math.ucla.edu/~asl/bsl/04-toc.htm
http://groups-beta.google.com/group/sci.math/msg/f0ab89956d4591f3
Dave L. Renfro
David C. Ullrich
wrote:
>I just accidentally learned that a paper relating to
>the topic of a thread I began on July 3, "MONOTONE
>SET FUNCTIONS, FIXED POINTS, AND CECH CLOSURE FUNCTIONS",
>is now available on-line: Preston C. Hammer's "General
>topology, symmetry, and convexity", Transactions of the
>Wisconsin Academy of Sciences, Arts and Letters 44 (1956),
>221-255.
>
>This got me thinking ... isn't this the journal that
>published William Dilworth's crankish math paper, "A
>correction in set theory" (MR 58 #16089)? Yep, it is.
>For those who don't know what I'm talking about, this
>is the crank paper mentioned in Underwood Dudley's 1992
>book "Mathematical Cranks" that disputes Cantor's proof
>of the uncountability of the reals and which managed
>to get published. Dudley's remarks caused him to be
>subsequently sued by Dilworth. For Usenet posts about
>this, see <http://tinyurl.com/cffrn>. 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.
>
>Circuit Opinion for Dilworth vs. Dudley:
>http://www.law.emory.edu/7circuit/jan96/95-2282.html
Well worth reading. My favorite bit:
'
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."
'
>On-line copy of William Dilworth's "A correction
>in set theory":
>http://digicoll.library.wisc.edu/cgi-bin/WI/WI-idx?type=article&byte=10235322
>
>An antidote to Dilworth's paper: Wilfrid Hodges,
>"An editor recalls some hopeless papers", Bulletin
>of Symbolic Logic 4 (1998), 1-16:
>http://www.math.ucla.edu/~asl/bsl/04-toc.htm
>http://groups-beta.google.com/group/sci.math/msg/f0ab89956d4591f3
>
>Dave L. Renfro
************************
David C. Ullrich
Jesse F. Hughes
> On 13 Jul 2005 18:14:17 -0700, "Dave L. Renfro" <renf...@cmich.edu>
> wrote:
[...]
>>
>>Circuit Opinion for Dilworth vs. Dudley:
>>http://www.law.emory.edu/7circuit/jan96/95-2282.html
>
> Well worth reading. My favorite bit:
>
> '
> 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."
> '
You *like* that section?
There goes about three-quarters of my plans for lawsuits.
Thank goodness that "half-wit" and "charlatan" aren't on that list. I
still have those to fall back on.
--
"For some reason mathematicians are pushing some weird mind-control
crap which works quite well from what I've seen over the last two
years, but I'm asking you to snap out of it!!!!"
-- James S. Harris learns the truth about the Mind Control Lasers
David C. Ullrich
<je...@phiwumbda.org> wrote:
>David C. Ullrich <ull...@math.okstate.edu> writes:
>
>> On 13 Jul 2005 18:14:17 -0700, "Dave L. Renfro" <renf...@cmich.edu>
>> wrote:
>
>[...]
>
>>>
>>>Circuit Opinion for Dilworth vs. Dudley:
>>>http://www.law.emory.edu/7circuit/jan96/95-2282.html
>>
>> Well worth reading. My favorite bit:
>>
>> '
>> 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."
>> '
>
>You *like* that section?
You'd like it too if you weren't a lazy, stupid,
crap-shooting, chicken-stealing idiot.
Yes, I like that section really a lot.
>There goes about three-quarters of my plans for lawsuits.
>
>Thank goodness that "half-wit" and "charlatan" aren't on that list. I
>still have those to fall back on.
************************
David C. Ullrich
Tony Orlow
Thought I'd take a look at this paper and see what he is correcting, in
particular. There are so many problems with Cantor! Well, one may call Dilworth
all sorts of things, but he is correct in his criticism of the diagonal proof
in terms of the number of permutations of digits and the fact that any complete
list of digital numbers is longer than it is wide. Too bad he was a little
rambling, and limited in scope. Thanks for the link. Crackpots unite! Well,
except for the ones that are actually nuts.....
>
> An antidote to Dilworth's paper: Wilfrid Hodges,
> "An editor recalls some hopeless papers", Bulletin
> of Symbolic Logic 4 (1998), 1-16:
> http://www.math.ucla.edu/~asl/bsl/04-toc.htm
> http://groups-beta.google.com/group/sci.math/msg/f0ab89956d4591f3
>
> Dave L. Renfro
>
>
--
Smiles,
Tony
Jim Spriggs
>
> ... the fact that any complete
> list of digital numbers is longer than it is wide.
Since there is no complete list (of the reals) it can be anything you
like.
--
I don't know who you are Sir, or where you come from,
but you've done me a power of good.
Dave L. Renfro
> This got me thinking ... isn't this the journal that
> published William Dilworth's crankish math paper, "A
> correction in set theory" (MR 58 #16089)? Yep, it is.
> For those who don't know what I'm talking about, this
> is the crank paper mentioned in Underwood Dudley's 1992
> book "Mathematical Cranks" that disputes Cantor's proof
> of the uncountability of the reals and which managed
> to get published. Dudley's remarks caused him to be
> subsequently sued by Dilworth. For Usenet posts about
> this, see <http://tinyurl.com/cffrn>. 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.
>
> Circuit Opinion for Dilworth vs. Dudley:
> http://www.law.emory.edu/7circuit/jan96/95-2282.html
>
> On-line copy of William Dilworth's "A correction
> in set theory":
> http://digicoll.library.wisc.edu/cgi-bin/WI/WI-idx?type=article&byte=10235322
>
> An antidote to Dilworth's paper: Wilfrid Hodges,
> "An editor recalls some hopeless papers", Bulletin
> of Symbolic Logic 4 (1998), 1-16:
> http://www.math.ucla.edu/~asl/bsl/04-toc.htm
> http://groups-beta.google.com/group/sci.math/msg/f0ab89956d4591f3
While I was packing some of my things yesterday I came
across my copy of Underwood Dudley's book. Because this
topic -- not accepting Cantor's diagonal proof for the
uncountability of the reals -- comes up so often in
sci.math, I thought I'd add to the archives an excerpt
from Dudley's book where he discusses Dilworth's paper.
Underwood Dudley, MATHEMATICAL CRANKS, The Mathematical
Association of America, 1992, x + 372 pages.
ISBN 0-88385-507-0
MR 93m:00003 The text of this review is at
http://www.math.niu.edu/~rusin/known-math/98/cranks
Also reviewed by Ian Stewart in American Mathematical
Monthly 101 (1994), 87-91.
>From pp. 44-45, the end of the chapter "Cantor's Diagonal
Process":
Another author, W.D., was not so radical as to deny
the existence of transfinite numbers; he merely refused
to accept Cantor's proof that there is more than one of
them. You do not expect to find Cantor refuted in the
pages of a scientific journal, but that is where D.
managed to have his 12-page paper, "A Correction in
Set Theory", published: in the Transactions of the [X.]
Academy of Sciences, Arts and Letters. "X" replaces the
name of one of our United States.
State academies of this sort, sometimes restricted to
science, sprung up in the nineteenth century and helped
sustain the intellectual life on the frontier. The
frontier of the intellectual life, that is, which moved
west much more slowly than the physical frontier. In
the nineteenth century they served a purpose, since
a college might have only one Professor of Natural
Philosophy, expected to provide instruction in physics,
chemistry, mathematics, and, from time to time, biology;
he might get lonely, and could use the chance to meet
with his colleagues once a year to find out the latest
developments in the teaching of science. Now state
academies have largely outlived their purpose, since
there are national and state organizations in each of
the scientific disciplines, and it is to the meetings
of those organizations that teachers go. But, as
institutions tend to do, they have lived on -- after all,
they do no one any harm and cost very little, so there
is no incentive for doing away with them -- but they no
longer attract material of the highest quality. In some
states, manuscripts are not sent to referees, and the
editor of the academy's Transactions, perhaps with an
assistant, decides to accept or reject a paper. There
is suspicion that in some states, _nothing_ is ever
rejected. This may be unfair to some disciplines, but
I think it is accurate for mathematics.
D. just could not bear the Banach-Tarski theorem: that
it is possible to take a sphere, divide it into a finite
number of pieces, and then reassemble the pieces to
form two spheres, each exactly as large as the original
one. This seems paradoxical, and so it would be if we
were talking about physical spheres made up of atoms.
But when we have spheres make up of uncountably many
points, each with no extension, the paradox disappears.
A line one inch long contains _exactly_ as many points
as a line two inches long. That does not bother me,
it does not bother any mathematician, students of
mathematics can be made to appreciate it, and its
paradox content is the same as in the Banach-Tarski
result. D. realized that the Banach-Tarski theorem
was a theorem of set theory, so if he could upset the
foundations of set theory, the Banach-Tarski theorem
would have to go.
To get rid of it, D. chose to prove that the set of
real numbers is countable, and that Cantor's diagonal
process is a snare and a delusion. This would eliminate
the Banach-Tarski problem from mathematics, along with
several other things. Stripped of its verbiage, D.'s
proof is the same erroneous one that bright undergraduates
sometimes make: D. argued (in effect; the verbiage makes
it not all that easy to see) that if we list the real
numbers between 0 and 1 as follows,
.1, .2, .3, .4, .5, .6, .7, .8, .9, .10, .11, .12, .13,
.14, ..., .99, .100, .101, ..., .999, .1000, ...,
putting down first the one-digit decimals, then the
two-digit ones, and so on, we will have a list of all
the real numbers between 0 and 1. The reply to this
argument -- which usually elicits an "Oh" after a few
seconds' thought from bright undergraduates -- that
the list contains only the terminating decimals and
none of the non-terminating ones, might not affect D.
at all. His article reads as if it is by someone
convinced, whose mind is not going to be changed
by anything. It is by, in two words, a crank, and
it is no credit to the state of X.
>From p. 354, "Notes" for the chapter "Cantor's Diagonal
Process":
The author of "A correction in set theory", in which
Cantor was refuted, was William Dilworth, whose article
appeared in the Transactions of the Wisconsin Academy
of Sciences, Arts and Letters in 1974.
Dave L. Renfro
Tony Orlow
> "Tony Orlow (aeo6)" wrote:
> >
> > ... the fact that any complete
> > list of digital numbers is longer than it is wide.
>
> Since there is no complete list (of the reals) it can be anything you
> like.
>
>
the digital number systems. According to the nature of such systems as sets of
all possible strings from a given alphabet, for digital numbers of a given
number of digits D, using a number base B, there are B^D possible values for
your digital string. For B greater than 1 and D greater than 0 (both integers,
obviously), B^D>D. Therefore, the number of strings in a non-empty list of
digital numbers of base greater than 1 is always greater than the number of
digits in each string, if the list contains all strings of that length. This
inequality does not diminish as D increases, but rather, increases
exponentially. The complete list of digital numbers is infinitely wide, but its
length is infinitely greater. After all, the length of the list is supposed to
be aleph_1, while the width is supposed to be aleph_0. if the length is
supposed to be aleph_0, then the width must be logB(aleph_0). It behooves us to
remember the properties of the elements with which we work, since any
construction or discussion of the infinite set depends on those properties. I
know this isn't the standard rhetoric on this but, oh well! I use non-standard
methods that work.
--
Smiles,
Tony
Tony Orlow
The author actually misrepresented the mirror of the natural numbers. It should
be .0, .1, .2, .3.....9, .01, .11, .21, .31,...., .91, .02, .....etc. The only
reason that the correspondence between these numbers and the naturals is
rejected is that, while mathematicians easily accept unending digits to the
right of the point, since they still represent finite quantities, they reject
the notion that there are infinite whole numbers, with infinite digits to the
left of the point. At the same time, they maintain that the set of natural
numbers is infinite, and yet, one cannot have an infinite set of unique strings
constructed from a finite alphabet, without having strings of infinite length
in the set, and such infinite digital numbers to the left of the point
represent infinite quantities. Therefore, your infinite set of natural numbers,
when properly allowed to include the infinite strings which it requires, is
exactly the mirror image of the digital representations of reals between 0 and
1.
Hope this answers your question. ;)
--
Smiles,
Tony
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
> Jim Spriggs said:
> > "Tony Orlow (aeo6)" wrote:
> > >
> > > ... the fact that any complete list of digital numbers is longer
> > > than it is wide.
> >
> > Since there is no complete list (of the reals) it can be anything
> > you like.
> >
> >
> Yeah, sure, if you want to entirely ignore the nature of, and rules
> regarding, the digital number systems.
Digital number systems are an add-on to the properties of reals. No
aximatization of or stanard model of the reals requiers them.
> According to the nature of
> such systems as sets of all possible strings from a given alphabet,
> for digital numbers of a given number of digits D, using a number
> base B, there are B^D possible values for your digital string. For B
> greater than 1 and D greater than 0 (both integers, obviously),
> B^D>D. Therefore, the number of strings in a non-empty list of
> digital numbers of base greater than 1 is always greater than the
> number of digits in each string, if the list contains all strings of
> that length. This inequality does not diminish as D increases, but
> rather, increases exponentially. The complete list of digital numbers
> is infinitely wide
The point is that no such "list" exists, if by "list" one means a
funcional image of the naturals.
> I use
> non-standard methods that work.
But only for TO, and only in theory.
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
Since no natural is more than one digit longer than its predecessor, and
we start with only one digit, at what point does TO suggest that adding
1 more digit gets us to more than finitely many digits?
> At the same time, they maintain that the set of natural
> numbers is infinite
At what point does TO suggest that adding one should be no longer
possible?
The set of naturals is unending and the number of digits allowed in a
natural representation is not bounded, but neither any one natural nor
any digit string is required to be other than finite in order to
accomplish this.
TO's quantifier dyslexia rears its ugly head again.
G. Frege
<ITSnetNOTcom#vir...@COMCAST.com> wrote:
>
> TO's quantifier dyslexia rears its ugly head again.
>
It seems that there is one thing most "anti-cantorians" have in common:
a general lack of mathematical abilities.
F.
Tony Orlow
Virgil makes up lies to defend his bullshit. Virgil, go fuck yourself.
--
Smiles,
Tony
G. Frege
wrote:
> > >
> > > TO's quantifier dyslexia rears its ugly head again.
> > >
> >
> > It seems that there is one thing most "anti-cantorians" have in common:
> > a general lack of mathematical abilities.
> >
> >
> There were no quantifier dyslexias there or anywhere else in anything I said.
> Virgil makes up lies to defend his bullshit. Virgil, go fuck yourself.
>
Quote from some other post/reply to TO:
>
> If you wrote these out as logical statements, you would see
> that you are mixing up the order of quantifiers:
>
> (1) forall b, exists s in S,
> (if b is a finite bound, then length(s) > b)
>
> (2) exists s in S, forall b
> (if b is a finite bound, then length(s) > b)
>
> Statement (1) says that the *set* S has no finite bound.
> Statement (2) says that S contains an *element* that has
> no finite bound. Those are two different statements.
>
>
> Daryl McCullough
> Ithaca, NY
>
F.
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
This is clearly the resonse of a person who has run out of valid
arguments.
Darko Aleksic
>Quote from some other post/reply to TO:
>
>>
>> If you wrote these out as logical statements, you would see
>> that you are mixing up the order of quantifiers:
>>
>> (1) forall b, exists s in S,
>> (if b is a finite bound, then length(s) > b)
>>
>> (2) exists s in S, forall b
>> (if b is a finite bound, then length(s) > b)
>>
>> Statement (1) says that the *set* S has no finite bound.
>> Statement (2) says that S contains an *element* that has
>> no finite bound. Those are two different statements.
I don't get it. I have no idea what you guys are talking about and I'm
a bit slow so - can someone run above quote sloooowly by me?
Darko