Obections to Cantor's Theory (Wikipedia article)
david petry
I'm in the process of writing an article about
objections to Cantor's Theory, which I plan to contribute
to the Wikipedia. I would be interested in having
some intelligent feedback. Here' the article so far.
***
While the pure mathematicians almost unanimously accept
Cantor's Theory (with the exception of a small group of
constructivists), there are lots of intelligent people who
believe it to be an absurdity. Typically, these people
are non-experts in pure mathematics, but they are people
who have who have found mathematics to be of great practical
value in science and technology, and who like to view
mathematics itself as a science.
These "anti-Cantorians" see an underlying reality to
mathematics, namely, computation. They tend to accept the
idea that the computer can be thought of as a microscope
into the world of computation, and mathematics is the
science which studies the phenomena observed through that
microscope. They claim that that paradigm includes all
of the mathematics which has the potential to be applied to
the task of understanding phenomena in the real world (e.g.
in science and engineering).
Cantor's Theory, if taken seriously, would lead us to believe
that while the collection of all objects in the world of
computation is a countable set, and while the collection of all
identifiable abstractions derived from the world of computation
is a countable set, there nevertheless "exist" uncountable sets,
implying (again, according to Cantor's logic) the "existence"
of a super-infinite fantasy world having no connection to the
underlying reality of mathematics. The anti-Cantorians see
such a belief as an absurdity (in the sense of being
disconnected from reality, rather than merely counter-intuitive).
The pure mathematicians tend to view mathematics as an art
form. They seek to create beautiful theories, which may happen
to be connected to reality, but only by accident. Those who apply
mathematics, tend to view mathematics as a science which explores
an objective reality (the world of computation). In science, truth
must have observable implications, and such a "reality check"
would reveal Cantor's Theory to be a pseudoscience; many of the
formal theorems in Cantor's Theory have no observable implications.
The artists see the requirement that mathematical statements must
have observable implications as a restriction on their intellectual
freedom.
The "anti-Cantorian" view has been around ever since Cantor
introduced his ideas. Witness the following quote from a
contemporary of Cantor:
"I don't know what predominates in Cantor's
theory - philosophy or theology, but I am sure
that there is no mathematics there" (Kronecker)
In the contemporary mainstream mathematical literature, there
is almost no debate over the validity of Cantor's Theory.
It was the advent of the internet which revealed just how
prevalent the anti-Cantorian view still is; there seems to be a
never-ending heated debate about Cantor's Theory in the Usenet
newsgroups sci.math and sci.logic. Typically, the
anti-Cantorians accuse the pure mathematicians of living in a
dream world, and the mathematicians respond by accusing the
anti-Cantorians of being imbeciles, idiots and crackpots.
It is plausible that in the future, mathematics will be split
into two disciplines - scientific mathematics (i.e. the science
of phenomena observable in the world of computation), and
philosophical mathematics, wherein Cantor's Theory is
merely one of the many possible "theories" of the infinite.
Stephen Montgomery-Smith
> I'm in the process of writing an article about
> objections to Cantor's Theory, which I plan to contribute
> to the Wikipedia. I would be interested in having
> some intelligent feedback. Here' the article so far.
I have to admit that I don't follow the anti-Cantorian arguments very
much, but when I do, I get the sense that they lack coherence, and
perhaps they lack even intellectual honesty.
I can see Kronecker's point of view, which I guess is that Cantor's
theories depends upon the existence of mathematical objects that don't
seem to exist in real life (e.g. what is a real number, really?). If
the anti-Cantorians argued at this level, I think that I would
essentially be in agreement with them. I also think that the
pro-Cantorians and anti-Cantorians could co-exist side by side, holding
different philosophies as to what mathematics represents, but agreeing
upon its practical consequences.
But I find that anti-Cantorians try to say something quite different,
which is that the Cantorian position is logically wrong. This is
clearly absurd, unless you change the laws of logic, and since they are
currently working well, and no-one is able to come up with something
different and sane, why change them?
I had this experience when I tried to enter into a discussion with an
anti-Cantorian about how perhaps the Cantor approach is helpful in
telling us that we don't need to be searching for a halting function,
since a Cantor/Turing style argumnt shows that they don't exist. But
the response I got from this person wasn't even wrong - it was shear
nonsense, and I quickly gave up.
Honestly, I feel that your article about anti-Cantorians is too generous
towards them, and in the final analysis I would not be supportive of
Wikipedia accepting such an article. I don't think that
anti-Cantorianism as I have experienced it is simply a different point
of view, rather I genuinely believe that those who propose such a
viewpoint are crackpots.
I hope that you are not yourself an anti-Cantorian whom I have
inadvertantly offended, or if you are I would certainly be interested in
hearing a non-crackpot approach against Cantor's arguments.
Best, Stephen
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
Seems fair enough to me, though it overlooks that what many
anti-Cantorians propose is as effectively counter-scientific as it is
counter-Cantor.
Are most anti-Cantorians actual scientists or are they, like WM, neither
mathematicians nor scientists.
Stephen J. Herschkorn
To which of the axioms of Zermelo-Frankel do anti-Cantorians object?
Can anti-Cantorians identify correctly a flaw in the proof that there
exists no enumeration of the subsets of the natural numbers?
Do anti-Cantorians accept that sum(i=1..infty, d_i / 10^i) "exists"
for each collection (d_i) of decimal digits (as i ranges over the
positive integers)? If so, how do they correctly justify the collection
of these real number are countable?
If one is to insert an entry on anti-Cantorians in any encylopedia, it
must include the answers or lack thereof to these questions.
--
Stephen J. Herschkorn sjher...@netscape.net
Math Tutor in Central New Jersey and Manhattan
Stephen Montgomery-Smith
> Cantor's Theory, if taken seriously, would lead us to believe
> that while the collection of all objects in the world of
> computation is a countable set, and while the collection of all
> identifiable abstractions derived from the world of computation
> is a countable set, there nevertheless "exist" uncountable sets,
> implying (again, according to Cantor's logic) the "existence"
> of a super-infinite fantasy world having no connection to the
> underlying reality of mathematics. The anti-Cantorians see
> such a belief as an absurdity (in the sense of being
> disconnected from reality, rather than merely counter-intuitive).
>
> The pure mathematicians tend to view mathematics as an art
> form. They seek to create beautiful theories, which may happen
> to be connected to reality, but only by accident. Those who apply
> mathematics, tend to view mathematics as a science which explores
> an objective reality (the world of computation). In science, truth
> must have observable implications, and such a "reality check"
> would reveal Cantor's Theory to be a pseudoscience; many of the
> formal theorems in Cantor's Theory have no observable implications.
> "I don't know what predominates in Cantor's
> theory - philosophy or theology, but I am sure
> that there is no mathematics there" (Kronecker)
If you are going to argue at this level, it seems to me that the problem
is not with the hierarchy of different super-infinities, but with the
very concept of infinity itself. So, for example, I am told that
Kronecker said something to the effect that "God invented the integers,
man invented the rest."
But I think that even Kronecker's statement is a huge statement of
faith. I would contend that if you stick to mathematics that is
actually observable in the real world, that even notions such as the set
of integers, or the principle of induction, are dreams invented by
mathematicians.
For example, consider the collection of all numbers between 1 and
googolplex. There are quite a large number of them (indeed most of
them) that one could never write down on a piece of paper - even if that
piece of paper was as large as the solar system. This means that a
bunch of these integers, for all practical purposes, simply don't exist.
And googolplex is small - what about googolplex taken to the power of
itself googolplex times?
Now you may counter that "in principle" we can write all these integers
down. But as I said, this is essentially an act of faith, which has no
visible proof. And if you can take this leap of faith, why not go a
step further, and believe in the set of all real numbers, which, if you
accept its existence, cannot logically be placed into one-to-one
correspondence with the integers. Indeed why not go the whole way, and
believe in the whole von-Neuman universe of sets? The only thing that
will stop you in this belief is if you find some mathematical
inconsistency. But as Goedel proved, this mathematical inconsistency
might even exist amongst regular old number theory - you will never know
unless you find it.
You could also argue that the set of integers is a convenient
abstraction to represent the notion of counting, which is definitely a
real life activity. (Of course, we never actually count up to
googolplex, so it really is an abstraction.) But then I would counter
that the real numbers are merely a convenient abstraction to represent
lengths and times and such like - definitely useful, because the
resulting theories like calculus have clear real world applications.
And if you accept this abstraction, even merely hypothetically, then you
must accept the correctness of the Cantor's diagonal argument.
Seriously, if you think that you have found a way to construct a one-one
correspondence between the integers and the real numbers, I strongly
advise you to spend a lot of time proof-reading your work. Because you
will not have only contradicted the Cantorians, but you will have
contradicted the whole way in which modern mathematicians think. I'm
not saying that it cannot be done, but so many people have tried
unsuccessfully that I am not going to spend a lot of time checking yet
another attempt.
Best, Stephen
mathman
Keith A. Lewis
>I can see Kronecker's point of view, which I guess is that Cantor's
>theories depends upon the existence of mathematical objects that don't
>seem to exist in real life (e.g. what is a real number, really?). If
>the anti-Cantorians argued at this level, I think that I would
>essentially be in agreement with them. I also think that the
>pro-Cantorians and anti-Cantorians could co-exist side by side, holding
>different philosophies as to what mathematics represents, but agreeing
>upon its practical consequences.
I agree with all that.
>But I find that anti-Cantorians try to say something quite different,
>which is that the Cantorian position is logically wrong. This is
>clearly absurd, unless you change the laws of logic, and since they are
>currently working well, and no-one is able to come up with something
>different and sane, why change them?
>
>I had this experience when I tried to enter into a discussion with an
>anti-Cantorian about how perhaps the Cantor approach is helpful in
>telling us that we don't need to be searching for a halting function,
>since a Cantor/Turing style argumnt shows that they don't exist. But
>the response I got from this person wasn't even wrong - it was shear
>nonsense, and I quickly gave up.
>
>Honestly, I feel that your article about anti-Cantorians is too generous
>towards them, and in the final analysis I would not be supportive of
>Wikipedia accepting such an article. I don't think that
>anti-Cantorianism as I have experienced it is simply a different point
>of view, rather I genuinely believe that those who propose such a
>viewpoint are crackpots.
There are several anti-Cantor cranks here who are very vocal. I think
your experience is skewed by that. Wikipedia articles which focus on cranks
would only encourage them. This article should be about Kronecker and his
ilk, not the nutjobs who disagree with Cantor because they lack the mental
capacity to understand him.
>I hope that you are not yourself an anti-Cantorian whom I have
>inadvertantly offended, or if you are I would certainly be interested in
>hearing a non-crackpot approach against Cantor's arguments.
As an "undecided" myself, I have three:
1. If a number is indescribeable and unrepresentable, it is irrelevant.
The set of relevant numbers has cardinality equal to the naturals.
2. No continuum has been discovered in physics -- everything seems to
change in finite units called quanta. That's the real world.
3. The Dedekind cut paradox. Rationals on [0,1] are totally ordered and
countable, yet there are an uncountable number of cuts based on this total
ordering. With any finite set of cardinality n, there are n+1 such cuts.
But N u {-1} is countable, so the set of Dedekind cuts should be.
I post my doubts because you asked. (And I have modified the subject line
to avoid pissing on David's thread.)
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
Lee Rudolph
>So, for example, I am told that
>Kronecker said something to the effect that "God invented the integers,
>man invented the rest."
>
>But I think that even Kronecker's statement is a huge statement of
>faith. I would contend that if you stick to mathematics that is
>actually observable in the real world, that even notions such as the set
>of integers, or the principle of induction, are dreams invented by
>mathematicians.
"If the integers did not exist, Man would have had to invent God."
--Peano applied to Kronecker
Lee Rudolph
Barb Knox
kle...@OMEGA.MITRE.ORG (Keith A. Lewis) wrote:
[snip]
>1. If a number is indescribeable and unrepresentable, it is irrelevant.
>The set of relevant numbers has cardinality equal to the naturals.
By that criterion, almost all natural numbers are irrelevant, since
there is clearly a finite limit to the Kolmogorov complexity of things
that can ever be described or represented in a finite universe.
It looks like you really should be taking an Ultrafinitist position on
this.
[snip]
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
quasi
And then there's this quote:
If math is ever proved inconsistent, we have too much invested --
we'll just change logic.
quasi
Robert Kolker
>
> It is plausible that in the future, mathematics will be split
> into two disciplines - scientific mathematics (i.e. the science
> of phenomena observable in the world of computation), and
> philosophical mathematics, wherein Cantor's Theory is
> merely one of the many possible "theories" of the infinite.
There is no call to put scare quotes around the word -theory-. The
theory of transfinite numbers is developed the same way as the theory of
algorithmically computable numbers. To wit, it is developed from
postulates by means of standard logical arguments. There is nothing
special about the logic used to prove theorems in the theory of
transfinite cardinals and ordinals.
The beef that the anti-cantorians have is with the axioms, not with the
means of deducing theorems from the axioms. The only constraint on
theories is that they be internally consistent. It is not necessary that
a mathematical theory be applicable to the physical world.
Bob Kolker
G. Frege
A. Lewis) wrote:
>
> 2. No continuum has been discovered in physics -- everything seems to
> change in finite units called quanta. That's the real world.
>
But mathematics is not interested in the real world. (If it were it
would be a natural science, which it is not.)
F.
Stephen Montgomery-Smith
> Stephen Montgomery-Smith <ste...@math.missouri.edu> writes in article <yUWCe.163211$x96.36939@attbi_s72> dated Mon, 18 Jul 2005 23:26:22 GMT:
>>I hope that you are not yourself an anti-Cantorian whom I have
>>inadvertantly offended, or if you are I would certainly be interested in
>>hearing a non-crackpot approach against Cantor's arguments.
>
>
> As an "undecided" myself, I have three:
>
> 1. If a number is indescribeable and unrepresentable, it is irrelevant.
> The set of relevant numbers has cardinality equal to the naturals.
Actually, what is a representable number? Are all the numbers between 1
and googolplex representable? In principle, yes? In practice,
certainly no.
And if I do accept your paradigm, I find it hard to ascribe any real
sense to your second statement. What does cardinality equal to the
naturals mean?
>
> 2. No continuum has been discovered in physics -- everything seems to
> change in finite units called quanta. That's the real world.
Yes, but those quanta seem to come in continuum like amounts - for
example, is the ratio of the different energy levels of an electron
necessarily a rational number? (I actually have no idea what the answer
is.)
> 3. The Dedekind cut paradox. Rationals on [0,1] are totally ordered and
> countable, yet there are an uncountable number of cuts based on this total
> ordering. With any finite set of cardinality n, there are n+1 such cuts.
> But N u {-1} is countable, so the set of Dedekind cuts should be.
Similarly all finite dimensional matrices have discrete spectra. But
infinite dimensional operators can have spectra that are even of the
second Baire category. There is a real problem with extrapolating
finitary ideas to the infinite. "Taking the limit" in a naive manner
doesn't always work.
My feelings are that there are genuine philosophical problems with the
current approach of modern mathematics. But my sense is that our
current ideas about philosophy are so primitive, that discussions that
try to resolve these issues can never rise above scholastic arguments
about how many angels can fit on a pinhead. Right now the current
approach to mathematics is extremely effective and useful. My
suggestion is that we wait another five hundred years before trying to
resolve these issues.
Stephen
The World Wide Wade
kle...@OMEGA.MITRE.ORG (Keith A. Lewis) wrote:
> >I hope that you are not yourself an anti-Cantorian whom I have
> >inadvertantly offended, or if you are I would certainly be interested in
> >hearing a non-crackpot approach against Cantor's arguments.
>
> As an "undecided" myself, I have three:
>
> 1. If a number is indescribeable and unrepresentable, it is irrelevant.
> The set of relevant numbers has cardinality equal to the naturals.
>
> 2. No continuum has been discovered in physics -- everything seems to
> change in finite units called quanta. That's the real world.
>
> 3. The Dedekind cut paradox. Rationals on [0,1] are totally ordered and
> countable, yet there are an uncountable number of cuts based on this total
> ordering. With any finite set of cardinality n, there are n+1 such cuts.
> But N u {-1} is countable, so the set of Dedekind cuts should be.
This really quite weak. 1. is incomprehensible without precise
definitions. Define "indescribeable" and "unrepresentable" and
"relevant" mathematically. 2. No equilateral triangle has been
discovered in physics either. The idea that mathematical concepts
are a subset of what has been discovered in physics is laughable.
3. There is no paradox. Infinite sets are different from finite
ones. Surprisingly different. You are confusing "surprising" with
paradox.
The World Wide Wade
<1121727755.1...@g44g2000cwa.googlegroups.com>,
"david petry" <david_lawr...@yahoo.com> wrote:
> These "anti-Cantorians" see an underlying reality to
> mathematics, namely, computation. They tend to accept the
> idea that the computer can be thought of as a microscope
> into the world of computation, and mathematics is the
> science which studies the phenomena observed through that
> microscope.
Such people are ignorant of mathematics. Why should we care what
they think?
Dave L. Renfro
>> 2. No continuum has been discovered in physics -- everything
>> seems to change in finite units called quanta. That's
>> the real world.
Stephen Montgomery-Smith wrote (in part):
> Yes, but those quanta seem to come in continuum like
> amounts - for example, is the ratio of the different
> energy levels of an electron necessarily a rational
> number? (I actually have no idea what the answer is.)
I think Keith needs to look back in his quantum
mechanics texts where bound energy states and
continuum energy states are discussed. Also,
all these wave functions and such are functions
of a _continuous_ variable (except maybe in
lattice gauge theories, Penrose's twisters, etc.,
of which I know nothing about).
Keith, how would we know if we even saw a continuum
in the real world? Personally, I haven't seen a 'two'
in the real world. Natural numbers, rationals, reals,
wave functions, distribution operators, etc. are tools
for our mind to understand and interpret the real world.
A question that can easily keep me awake at night is
whether it's possible for an alien intelligence to be
so different from us (and I mean _really_ different,
such as not even being matter-based, but perhaps
existing as ripples in space-time or something else
way different from us) that it can develop an ability
to understand and interpret aspects of reality using
methods that we can't imagine (not intuitive-based,
not logical/temporal mathematical based, but something
unreachable by us). On the flip side, even the idea of
counting things might be utterly foreign to how they
experience reality, to say nothing of the rest of
mathematics. (Actually, I think I've had some students
that you could almost say this about!)
I guess it comes down to what "intelligent" means.
Is a river intelligent because of the huge number of
work-energy calculations taking place in order to
minimize its Hamiltonian as it flows over and around
things? Even a grain of sand falling a centimeter in
the air is, in a sense, performing more "calculations
with the universe" than all the calculators, computers,
and supercomputers together have performed for mankind.
Dave L. Renfro
quasi
<david_lawr...@yahoo.com> wrote:
>
>I'm in the process of writing an article about
>objections to Cantor's Theory, which I plan to contribute
>to the Wikipedia. I would be interested in having
>some intelligent feedback. Here' the article so far.
>
> ....
>
>These "anti-Cantorians" see an underlying reality to
>mathematics, namely, computation. They tend to accept the
>idea that the computer can be thought of as a microscope
>into the world of computation, and mathematics is the
>science which studies the phenomena observed through that
>microscope. They claim that that paradigm includes all
>of the mathematics which has the potential to be applied to
>the task of understanding phenomena in the real world (e.g.
>in science and engineering).
To me, it sounds very presumptuous to talk about anti-Cantorians as if
they were a well defined group. Much of what you say strikes me as
your own opinion. essentially an editorial, but camouflaged by
attributing the views to a group. If you could even assemble a group
of anti-Cantorians -- try it, I dare you -- I'll bet they would
disagree with each other on almost everything.
quasi
Helene....@wanadoo.fr
Barb Knox wrote:
> In article <dbhhdm$flk$1...@newslocal.mitre.org>,
> kle...@OMEGA.MITRE.ORG (Keith A. Lewis) wrote:
> [snip]
>
> >1. If a number is indescribeable and unrepresentable, it is irrelevant.
> >The set of relevant numbers has cardinality equal to the naturals.
>
> By that criterion, almost all natural numbers are irrelevant, since
> there is clearly a finite limit to the Kolmogorov complexity of things
> that can ever be described or represented in a finite universe.
>
> It looks like you really should be taking an Ultrafinitist position on
> this.
>
Just to point out that Cantor's Theorem turns on the existence of
types, not on infinitary reasoning. Suppose there is a maximum natural
number n. A real number is one type up, namely a function R from the
natural numbers to the natural numbers, satisfying certain conditions.
Remark that when we say "natural numbers" here we mean {0,1,2,...,n},
so a real number is a function from {0,1,...,n} to {0,1,...n}
satisfying certain conditions. For instance, if we're considering real
numbers in base 10, then R(0) must be a natural number and, for i > 0,
R(i) must be in the set {0,1,2,...9}. One can embed the natural
numbers in the set of all such R (the natural numbers are the functions
where R(i) = 0 for all i > 0).
In this ultrafinite world are there more reals than natural numbers?
Yes. And one can use the diagonal argument to construct it.
David Kastrup
> In article <1121727755.1...@g44g2000cwa.googlegroups.com>,
> "david petry" <david_lawr...@yahoo.com> wrote:
>
>> I'm in the process of writing an article about
>> objections to Cantor's Theory, which I plan to contribute
>> to the Wikipedia. I would be interested in having
>> some intelligent feedback. Here' the article so far.
>
> Seems fair enough to me, though it overlooks that what many
> anti-Cantorians propose is as effectively counter-scientific as it
> is counter-Cantor.
It seems completely unfair and absurd to me since it gives the
"Anti-Cantorians" equal consideration/hearing. That overlooks that
a) it is a minuscule ratio of people
b) they are visible only in non-professional circles like Usenet
c) none of those that are visible can sustain a coherent argument or
present a more or less consistent case. In fact, the ratio of
people among them that are unable to understand the meaning of the
order of quantifiers is almost 100%.
d) there is no consistent theory that is put forward instead of what
they aim to replace. Every crank has his own, different version of
lala land that he presents.
I won't say that serious mathematical philosophy that would be
incompatible with findings from Cantor is impossible, and it would
likely have to be based on different axioms. But if somebody came up
with it, he has found no capable voice presenting him, at least here.
Here we just have a small number of mathematically incapable but very
verbose cranks that all have their own pet theories that are
self-inconsistent at the most basic levels, let alone incompatible
with each other.
Giving them such consideration as in your "article" is completely
lunatic. Their criticism boils down to "math must not be hard for
me." We don't a Handicapper General in math:
<URL:http://www.tnellen.com/cybereng/harrison.html>
It is very much inappropriate to give them the exposure they don't get
elsewhere.
The article is much too long. It can be subsumed into
Cantor's work leads to quite unintuitive results, while still
being quite accessible to the layman. It has met opposition from
mathematically competent opponents at its time but has, partly
connected with changes in set theory, been made an integral part
of today's mathematics. While "anti-Cantorians" make themselves
quite visible on Usenet groups, they are actually few but
prolific, with a non-mathematical background, and unable to put
forward a coherent argument. Remarkably prevalent among them is
the inability to understand nested quantifiers.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
Helene....@wanadoo.fr
david petry wrote:
<snip>
There is no mention of one historical or living figure who is
anti-Cantorian, what their objections were, and what logical and
mathematical systems they advocated. It's a rant. Maybe Wikipedia
accepts rants, but if it does it will quickly lose any status it might
have (does it have any?).
Helene....@wanadoo.fr
David Kastrup wrote:
> It seems completely unfair and absurd to me since it gives the
> "Anti-Cantorians" equal consideration/hearing. That overlooks that
>
> a) it is a minuscule ratio of people
> b) they are visible only in non-professional circles like Usenet
> c) none of those that are visible can sustain a coherent argument or
> present a more or less consistent case. In fact, the ratio of
> people among them that are unable to understand the meaning of the
> order of quantifiers is almost 100%.
> d) there is no consistent theory that is put forward instead of what
> they aim to replace. Every crank has his own, different version of
> lala land that he presents.
>
> I won't say that serious mathematical philosophy that would be
> incompatible with findings from Cantor is impossible, and it would
> likely have to be based on different axioms. But if somebody came up
> with it, he has found no capable voice presenting him, at least here.
>
Solomon Feferman is probably the best known living anti-Cantorian.
Chan-Ho Suh
david petry <david_lawr...@yahoo.com> wrote:
> I'm in the process of writing an article about
> objections to Cantor's Theory, which I plan to contribute
> to the Wikipedia. I would be interested in having
> some intelligent feedback. Here' the article so far.
>
The truth of the matter is that the article you wrote constitutes
*original research* on your part, despite your attempt to ascribe your
views to the "anti-Cantorians", which is, as quasi pointed out, not a
well-defined group. Thus, it is not acceptable for inclusion into
Wikipedia. You should read more on Wikipedia policy before wasting
everyone's time including your own.
Your article will either be VFD'd or successively modified in such a
manner that you will be greatly unhappy with the result. But before
any of that happens, you will have wasted a lot of other people's time
and energy.
Jesse F. Hughes
Exactly.
The quoted paragraph gives it away. This isn't about
"anti-Cantorians", whatever the hell that might mean. This is about
Petrians, a well-defined group in which there is no dissension at all
(since there is only one member).
Well-put.
--
"I am one of the more important discoverers in mathematical history,
but future students will have the luxury of knowing that, and may be
puzzled by your behavior now." -- James Harris
(At least I have the foresight to quote his pearls of wisdom.)
Jesse F. Hughes
to give legitimacy to your own crankish notions.
> there are lots of intelligent people who believe [Cantor's Theory]
> to be an absurdity.
No editorializing there, huh? Anyway, what is "Cantor's Theory"?
> These "anti-Cantorians" see an underlying reality to mathematics,
> namely, computation.
Petry's own particular nonsense and not a broad program.
> Cantor's Theory, if taken seriously,[...]
Uh-huh. What is this theory that intelligent people don't take
seriously, anyway?
> In science, truth must have observable implications, and such a
> "reality check" would reveal Cantor's Theory to be a pseudoscience
> [...]
Science does not proclaim that all truth has observable implications.
You are confusing science with particular forms of empiricism: a
philosophical, not a scientific, position.
In any case, Cantor's Theory (*what* is it, again?) can be
pseudoscience only if it is a scientific theory. Perhaps you can
defend this claim once you tell us what Cantor's Theory is.
> The artists see the requirement that mathematical statements must
> have observable implications as a restriction on their intellectual
> freedom.
No editorializing in that excerpt, huh?
> It was the advent of the internet which revealed just how prevalent
> the anti-Cantorian view still is [...]
It was the advent of the internet that let Petry complain a bit more
vocally about his own particular delusions.
He was shocked to discover others had delusions that shared a
consequence: Cantor is a bad, bad man.
> It is plausible that in the future, mathematics will be split
> into two disciplines - scientific mathematics (i.e. the science
> of phenomena observable in the world of computation), and
> philosophical mathematics, wherein Cantor's Theory is
> merely one of the many possible "theories" of the infinite.
Bullhonkies. It is plausible only in Petry's feeble imagination.
This text is utterly unworthy of Wikipedia. This is persuasion, not
exposition, and it attempts to persuade that philosophically naive
views present a proper controversy. Hogwash.
But if you want to gussy this up to fool the naive, the least you
could do is define "Cantor's Theory". It doesn't appear in any
literature of which I am aware.
--
"Puts his arm around you, fiddles with your hair. You know, and he says, come
on, you know, just because you like a bit of a kiss and a cuddle with another
man doesn't make you gay. Which, you know, I've thought a lot about. But I
think it does. I think it does." --- The Office (interviews)
Jesse F. Hughes
I don't know about status, but I find Wikipedia useful on many
occasions.
Of course, one must always take its articles with a grain of salt.
--
Jesse F. Hughes
"Now 'pure math' makes sense as well as clearly it's a peacock game,
where some of you see it as a way to show you as being highly
intelligent and thus more desirable to women." -- James S. Harris
Alec McKenzie
> Can anti-Cantorians identify correctly a flaw in the proof that there
> exists no enumeration of the subsets of the natural numbers?
In my view the answer to that question a definite "No, they
can't".
However, the fact that no flaw has yet been correctly identified
does not lead to a certainty that such a flaw cannot exist. Yet
that is just what pro-Cantorians appear to be asserting, with no
justification that I can see.
--
Alec McKenzie
mcke...@despammed.com
Barb Knox
Alec McKenzie <mcke...@despammed.com> wrote:
> "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
>
>> Can anti-Cantorians identify correctly a flaw in the proof that there
>> exists no enumeration of the subsets of the natural numbers?
>
>In my view the answer to that question a definite "No, they
>can't".
>
>However, the fact that no flaw has yet been correctly identified
>does not lead to a certainty that such a flaw cannot exist.
Actually, in this case it does. The proof is simple enough to be
thoroughly check by humans and computers. There is no flaw in the proof.
>Yet
>that is just what pro-Cantorians appear to be asserting, with no
>justification that I can see.
Do you dispute the above justification?
David Kastrup
> "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
>
>> Can anti-Cantorians identify correctly a flaw in the proof that
>> there exists no enumeration of the subsets of the natural numbers?
>
> In my view the answer to that question a definite "No, they can't".
>
> However, the fact that no flaw has yet been correctly identified
> does not lead to a certainty that such a flaw cannot exist.
Uh, what? There is nothing fuzzy about the proof.
Suppose that a mapping of naturals to the subsets of naturals exists.
Then consider the set of all naturals that are not member of the
subset which they map to.
The membership of each natural can be clearly established from the
mapping, and it is clearly different from the membership of the
mapping indicated by the natural. So the assumption of a complete
mapping was invalid.
> Yet that is just what pro-Cantorians appear to be asserting, with no
> justification that I can see.
Uh, where is there any room for doubt? What more justification do you
need apart from a clear 7-line proof? It simply does not get better
than that.
Jesse F. Hughes
Huh?
The proof of Cantor's theorem is easily formalized. It's remarkably
short and simple and every step can be verified as correct.
It is perfectly reasonable to assert that no such flaw exists (given
the axioms used in the proof). Indeed, why would anyone entertain any
doubts when he can confirm the correctness of each and every step of
the proof?
--
Jesse F. Hughes
"How come there's still apes running around loose and there are
humans? Why did some of them decide to evolve and some did not? Did
they choose to stay as a monkey or what?" -Kans. Board of Ed member
Han de Bruijn
> 2. No continuum has been discovered in physics -- everything seems to
> change in finite units called quanta. That's the real world.
I quote this from an old poster (Re: Cantor's diagonal proof wrong?):
> It's evident from your argument that you have not a well founded idea
> about Applied Science and how it works. If you think that "continuity"
> means "not discrete" there, then you are simply mistaken. For example,
> in fluid dynamics everything is built up from molecules. Yet everybody
> in that discipline is working with partial differential equations, as
> if the discrete substrate didn't exist at all! The secret behind this
> is the phenomenon that a continuum, in physics, is characterized by the
> fact that its numbers are inaccurate, thus blurring everything which
> is below a certain level of perception.
>
> One of the most striking examples in this context has been the discovery
> of the so-called Fluid-Tube Continuum. The idea behind this comes from
> the classical theory of Porous Media. It is virtually impossible, namely
> to apply the original Navier-Stokes / Heat Transfer equations, together
> with their boundary conditions, to a truly detailed model of the tubes
> in a heat exchanger. With help of the porous media theory, though, tube
> bundles become amenable to mathematical treatment. The trick is that all
> tubes are "blurred" in such a way that they aren't even visible anymore.
>
> Resulting in truly continuous models for the governing equations, which
> actually become partial differential equations, as usual. Yeah, and then
> the numerical method comes in. And the continuized PD equations must be
> discretized again. If you are interested, here is more about it:
>
> http://huizen.dto.tudelft.nl/deBruijn/grondig/science.htm#cz
Han de Bruijn
Han de Bruijn
> [ ... snip ... ] There is a real problem with extrapolating
> finitary ideas to the infinite. "Taking the limit" in a naive manner
> doesn't always work.
You say "naive". That intrigues me. Would you say that taking the limit
in a "less naive" manner, instead, would always work ?
> [ ... snip ... ] Right now the current
> approach to mathematics is extremely effective and useful.
I will not deny that it is effective. But I don't find it so _extremely_
effective. I don't think that _efficiency_ is a hot item in theoretical
mathematics. Has it ever been ?
Han de Bruijn
Han de Bruijn
> [ ... snip ... ] 2. No equilateral triangle has been
> discovered in physics either. The idea that mathematical concepts
> are a subset of what has been discovered in physics is laughable.
But the _idea_ of an equilateral triangle _has_ been discovered in
physics. The idea that mathematical concepts are _idealizations_ of
what has been discovered in physics is _not_ laughable.
Nobody seems to comprehend, though, that _idealizations_ may provide
the clue to understanding: where the conflict between "realists" and
"theorists" comes from. And how it can finally be solved.
Han de Bruijn
Han de Bruijn
> Cantor's work leads to quite unintuitive results, while still
> being quite accessible to the layman. It has met opposition from
> mathematically competent opponents at its time but has, partly
> connected with changes in set theory, been made an integral part
> of today's mathematics. While "anti-Cantorians" make themselves
> quite visible on Usenet groups, they are actually few but
> prolific, with a non-mathematical background, and unable to put
> forward a coherent argument. Remarkably prevalent among them is
> the inability to understand nested quantifiers.
Non-mathematical background ? Look at yourself ! You claim that you know
something about Numerical Analysis, while it is quite clear from your
postings that you don't even have a clue.
Han de Bruijn
Dave Seaman
You, on the other hand, have shown that you do not understand the
difference between numerical analysis and mere numerical methods. Hint:
the former includes error analysis.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
David Kastrup
Actually, I did some course work during my diploma studies. And
helped out on the diploma thesis of a befriended mathematician that
was trying to approximate some recursively defined probability
distributions of piecewise exponential characteristics, proving to her
that her naive approach of Simpson's rule was leading to large
cascading errors (as well as exponentially increasing runtime). She
was not alone with that ill-considered approach: the distribution in
question had been handled that way in literature, taking weeks of
computation time and coming up basically with junk. And yes,
numerical analysis would not have just provided the method (which was
employed here), but also the error estimates. And ignoring them was
what turned this from mathematics into hand-waving.
So we figured out how to do this semi-symbolically (this was before
the widespread advent of symbolic calculation) by using combined
exponentials and polynomials, and _useful_ results (namely with
controllable errors) dropped out after few minutes of runtime (the
symbolic expressions had a few hundred terms, reasonably fast to
evaluate, but infeasible for manual calculation).
In my engineering studies and diploma thesis, I also had to work a lot
with error propagation in sliding-window Fourier transforms and
synthesis, and I also did quite a bit of fixed and floating point
arithmetic applications as a programmer where error estimates were
important.
sradhakr
Dear Prof. Montgomery-Smith,
The Cantorian viewpoint has been seriously challenged by my newly
proposed logic NAFL (Int. J. Quant. Inf., vol. 3, No. 1 (2005), pp.
263-267; see also <http://philsci-archive.pitt.edu/archive/00001923/>,
math.LO/0506475, cs.LO/0411094, quant-ph/0504115. Please see my
responses below to your message.
Stephen Montgomery-Smith wrote:
> david petry wrote:
> > I'm in the process of writing an article about
> > objections to Cantor's Theory, which I plan to contribute
> > to the Wikipedia. I would be interested in having
> > some intelligent feedback. Here' the article so far.
>
>
> I have to admit that I don't follow the anti-Cantorian arguments very
> much, but when I do, I get the sense that they lack coherence, and
> perhaps they lack even intellectual honesty.
I submit that it is the Cantorians who are intellectually dishonest,
for they have failed to respond to my arguments presented in the above
REFEREED, PUBLISHED work. For that matter, I don't think that refereed
publications are essential in this day and age of instant electronic
communication. If significant claims are posted to respectable
electronic archives like the arXiv and the PhilSci Archive, then the
academic community has a duty to respond, if only to correct any
misunderstandings that may arise due to the wide reach of these
archives.
>
> I can see Kronecker's point of view, which I guess is that Cantor's
> theories depends upon the existence of mathematical objects that don't
> seem to exist in real life (e.g. what is a real number, really?). If
> the anti-Cantorians argued at this level, I think that I would
> essentially be in agreement with them. I also think that the
> pro-Cantorians and anti-Cantorians could co-exist side by side, holding
> different philosophies as to what mathematics represents, but agreeing
> upon its practical consequences.
>
> But I find that anti-Cantorians try to say something quite different,
> which is that the Cantorian position is logically wrong. This is
> clearly absurd, unless you change the laws of logic, and since they are
> currently working well, and no-one is able to come up with something
> different and sane, why change them?
>
> I had this experience when I tried to enter into a discussion with an
> anti-Cantorian about how perhaps the Cantor approach is helpful in
> telling us that we don't need to be searching for a halting function,
> since a Cantor/Turing style argument shows that they don't exist. But
> the response I got from this person wasn't even wrong - it was shear
> nonsense, and I quickly gave up.
>
> Honestly, I feel that your article about anti-Cantorians is too generous
> towards them, and in the final analysis I would not be supportive of
> Wikipedia accepting such an article. I don't think that
> anti-Cantorianism as I have experienced it is simply a different point
> of view, rather I genuinely believe that those who propose such a
> viewpoint are crackpots.
>
> I hope that you are not yourself an anti-Cantorian whom I have
> inadvertently offended, or if you are I would certainly be interested in
> hearing a non-crackpot approach against Cantor's arguments.
>
> Best, Stephen
>
The logic NAFL is the correct approach, and not just against Cantor's
arguments; NAFL has important positive aspects, as is clear from my
papers. So far the international academic community has pretended that
NAFL does not exist, rather than answering me point by point. How does
one deal with people who fail to acknowledge the existence of something
as important as a new logic, a new philosophy of mathematical truth, a
new way of doing theoretical science (physics, mathematics, computer
science), etc.? I used to think that maybe it is my fault, maybe NAFL
isn't that important, maybe I haven't explained my ideas clearly, etc.
But not any more. Now that my work has been published, the onus is on
the academic community to honestly evaluate/criticize it and give NAFL
its due. I am assuming, of course, that the academic community consists
of honest, sincere people who are genuinely interested in taking
science forward, rather than merely "protecting turf".
Regards, R. Srinivasan
David C. Ullrich
I once had a person tell me the following, with a straight face:
(*) "You can't say for sure there's no such thing as a square
circle! I mean just because they haven't found one yet doesn't
mean they won't discover one tomorrow."
Please choose one of the following replies:
(i) No, (*) is nonsense. If it's square then _by definition_
it's not a circle. So they will _never_ find a square circle.
(ii) Hmm, good point.
You really should choose one of (i) or (ii), so people know
how to reply to your post. The point: If you say (ii) then
we know that there's no point worrying about anything you
say. Otoh if you say (i) then there's hope - you agree that
we're _certain_ they will never find a square circle, now
we just have to convince you that our assertions about
enumerations of subsets of N are just as certain, for
entirely similar (although slightly more complicated)
reasons.
So which is it, (i) or (ii?
************************
David C. Ullrich
tor...@sm.luth.se
david petry wrote:
> I'm in the process of writing an article about
> objections to Cantor's Theory, which I plan to contribute
> to the Wikipedia. I would be interested in having
> some intelligent feedback.
It's open to anybody to introduce any kind of rant into
the Wikipedia. Go for it!
Han de Bruijn
> You, on the other hand, have shown that you do not understand the
> difference between numerical analysis and mere numerical methods. Hint:
> the former includes error analysis.
OK, then my expertise is in numerical methods only. Who cares ...
On the other hand, I find that the vendors of those big Finite Element
packages do call their products NA, not NM. So ...
Han de Bruijn
Stephen Montgomery-Smith
> Stephen Montgomery-Smith wrote:
>
>> [ ... snip ... ] There is a real problem with extrapolating finitary
>> ideas to the infinite. "Taking the limit" in a naive manner doesn't
>> always work.
>
>
> You say "naive". That intrigues me. Would you say that taking the limit
> in a "less naive" manner, instead, would always work ?
Interesting question. In think deep in my heart I do believe this. But
the evidence is against me. E.g. how do you get type III von-Neuman
algebras as the limit of f.d. matrix algebras?
>> [ ... snip ... ] Right now the current approach to mathematics is
>> extremely effective and useful.
>
>
> I will not deny that it is effective. But I don't find it so _extremely_
> effective. I don't think that _efficiency_ is a hot item in theoretical
> mathematics. Has it ever been ?
OK, I'll drop the superlative.
Han de Bruijn
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
>> Non-mathematical background ? Look at yourself ! You claim that you know
>> something about Numerical Analysis, while it is quite clear from your
>> postings that you don't even have a clue.
> [ ... valid counter argument snipped ... ]
Allright. Fair enough. I apologize for having responded too quickly,
without knowing some facts. On the other hand, you must realize that
such error analysis may become rather clueless with for example F.E.
analysis in i.e. a Civil Engineering application. I don't want to say
that your work has been relatively "easy", but it has been different
from that "heavy" kind of number crunching.
On the other hand, you shouldn't deny any mathematical background to
everybody who disagrees with you on those Cantor issues.
Han de Bruijn
Alec McKenzie
It is (i), of course. But you seem to be suggesting that the
proof in question is flawless for similar reasons to its being
so _by definition_. That I cannot see.
--
Alec McKenzie
mcke...@despammed.com
Jesse F. Hughes
> It's open to anybody to introduce any kind of rant into
> the Wikipedia. Go for it!
Really, Torkel, whether you are impressed by Wikipedia or not, you
shouldn't encourage folks to deface the site.
--
Jesse F. Hughes
"Surround sound is going to be increasingly important in future
offices."
-- Microsoft marketing manager displays his keen insight
Alec McKenzie
> Alec McKenzie <mcke...@despammed.com> writes:
>
> > "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
> >
> >> Can anti-Cantorians identify correctly a flaw in the proof that
> >> there exists no enumeration of the subsets of the natural numbers?
> >
> > In my view the answer to that question a definite "No, they can't".
> >
> > However, the fact that no flaw has yet been correctly identified
> > does not lead to a certainty that such a flaw cannot exist.
>
> Uh, what? There is nothing fuzzy about the proof.
I am not suggesting there is anything fuzzy about the proof.
> Suppose that a mapping of naturals to the subsets of naturals exists.
> Then consider the set of all naturals that are not member of the
> subset which they map to.
>
> The membership of each natural can be clearly established from the
> mapping, and it is clearly different from the membership of the
> mapping indicated by the natural. So the assumption of a complete
> mapping was invalid.
>
> > Yet that is just what pro-Cantorians appear to be asserting, with no
> > justification that I can see.
>
> Uh, where is there any room for doubt? What more justification do you
> need apart from a clear 7-line proof? It simply does not get better
> than that.
I quite agree that it does not get better than that, but I think
one must allow some room for doubt, however small, for any
proof. Otherwise one is proclaiming infallibility.
It has been known for a proof to be put forward, and fully
accepted by the mathematical community, with a fatal flaw only
spotted years later.
--
Alec McKenzie
mcke...@despammed.com
tor...@sm.luth.se
Jesse F. Hughes wrote:
> Really, Torkel, whether you are impressed by Wikipedia or not, you
> shouldn't encourage folks to deface the site.
Well, rants are unlikely to survive long in Wikipedia. So
if somebody asks for feedback on a proposed rant, surely the
simple and reasonable course is simply to encourage him to
post it.
David Kastrup
> David Kastrup <d...@gnu.org> wrote:
>> Alec McKenzie <mcke...@despammed.com> writes:
>>
>> > However, the fact that no flaw has yet been correctly identified
>> > does not lead to a certainty that such a flaw cannot exist.
>>
>> Uh, what? There is nothing fuzzy about the proof.
>
> I am not suggesting there is anything fuzzy about the proof.
>
>> Suppose that a mapping of naturals to the subsets of naturals exists.
>> Then consider the set of all naturals that are not member of the
>> subset which they map to.
>>
>> The membership of each natural can be clearly established from the
>> mapping, and it is clearly different from the membership of the
>> mapping indicated by the natural. So the assumption of a complete
>> mapping was invalid.
>>
>> > Yet that is just what pro-Cantorians appear to be asserting, with
>> > no justification that I can see.
>>
>> Uh, where is there any room for doubt? What more justification do
>> you need apart from a clear 7-line proof? It simply does not get
>> better than that.
>
> I quite agree that it does not get better than that, but I think one
> must allow some room for doubt, however small, for any
> proof.
Well, it is now about a hundred years later, and millions of
mathematicians have checked those 7 lines. It is not like they are
difficult to comprehend or something.
> Otherwise one is proclaiming infallibility.
>
> It has been known for a proof to be put forward, and fully accepted
> by the mathematical community, with a fatal flaw only spotted years
> later.
In a concise 7 line proof? Bloody likely. And that's what you call
"with no justification that I can see".
I mean, look up "justification" in a dictionary of your choice. It
would be hard to find anything _more_ justified.
David C. Ullrich
<mcke...@despammed.com> wrote:
Fabulous.
>But you seem to be suggesting that the
>proof in question is flawless for similar reasons to its being
>so _by definition_. That I cannot see.
Ok, here's another question. Suppose that we want to
prove that A implies B. Suppose that we have an
completely flawless proof that A implies C, and
a completely flawless proof that C implies B.
Does the union of those two proofs constitute
a flawless proof that A implies B?
I imagine you'll say yes to that as well. But
the proof of the theorem in question really
does involve nothing more than statements
which are true by definition and statements
which follow from previous statements by
"if A implies C and C implies B then A implies
B" arguments.
************************
David C. Ullrich
Jesse F. Hughes
It still seems like you're inviting work for the Wikipedia editors.
Just as a matter of kindness to them, I prefer to discourage Petry.
--
Jesse F. Hughes
"You know that view most people have of mathematicians as brilliant
people? What if they're not?" -- James S. Harris
Jesse F. Hughes
> Ok, here's another question. Suppose that we want to
> prove that A implies B. Suppose that we have an
> completely flawless proof that A implies C, and
> a completely flawless proof that C implies B.
> Does the union of those two proofs constitute
> a flawless proof that A implies B?
Am I the only one suffering from flashbacks of Achilles and the
Tortoise?
Fortunately published back when copyrights really did expire.
<http://www.ditext.com/carroll/tortoise.html>
--
"I am the barbarian at the gates, raw creative force, willpower, and
the will to fight for the truth no matter what, no matter who stands
against me, no matter how many of you band [...] together in your
weakness to fight against the math." -- James S. Harris
Alec McKenzie
> Alec McKenzie <mcke...@despammed.com> writes:
> > It has been known for a proof to be put forward, and fully accepted
> > by the mathematical community, with a fatal flaw only spotted years
> > later.
>
> In a concise 7 line proof? Bloody likely.
I doubt it had seven lines, but I really don't know how many.
Probably many more than seven.
> And that's what you call "with no justification that I can see".
No, it is not -- what I called "with no justification that I can
see" was something else:
It was the assertion that no flaw having been found in a proof
leads to a certainty that such a flaw cannot exist. I still see
no justification for that.
--
Alec McKenzie
mcke...@despammed.com
Ross A. Finlayson
In one sense that's constructively illustrable via induction, with
well-ordering and transfer. That's basically a three line proof that
infinite sets are equivalent.
Another notion is that it leads to the conclusion of the dual
representation of an ur-element as the null element and the universal
element.
In a von Neumann universe, or rather ubiquitous ordinals, the powerset
is successor is order type. In the bijection f(x)=x+1, S(N)={},
S({})=N, and the direct sum of infinitely many copies of N is the empty
set.
With regards to the naturals and a segment of the reals, nested
intervals reinforces the notion that a well-ordering of the reals shows
the structure of the reals to be a contiguous sequence of points.
It's "Post-Cantorian."
Particularly with regards to bijections between the naturals and reals,
there are a variety of useful analytical results that do exist.
Consider Vitali's reasoning why there exist unmeasurable sets, and why
the existence of variously an infinitesimal or double infinitesimal, or
"continuous" and "discrete" infinitesimal, on a one-dimensional line,
leads to a ready explanation why it is not so.
This way of reasoning leads to perhaps more applicable results than the
transfinite cardinals, which in general have no utility.
Skolemize, your model is countable. Via induction, the order type of
all ordinals would be an ordinal. Quantify.
V = L anyways. Appealing to constructibility, or finitudinous, finity,
does not avoid variously these consequences, with infinity, and the
infinite universe is the infinite constructible universe.
Obviously I suggest the null axiom theory. ZF is inconsistent.
Ross
--
"Also, consider this: the unit impulse function times
one less twice the unit step function times plus/minus
one is the mother of all wavelets."
David Kastrup
> David Kastrup <d...@gnu.org> wrote:
>
>> Alec McKenzie <mcke...@despammed.com> writes:
>> > It has been known for a proof to be put forward, and fully accepted
>> > by the mathematical community, with a fatal flaw only spotted years
>> > later.
>>
>> In a concise 7 line proof? Bloody likely.
>
> I doubt it had seven lines, but I really don't know how many.
> Probably many more than seven.
It was seven lines in my posting. You probably skipped over it. It
is a really simple and concise proof. Here it is again, for the
reading impaired, this time with a bit less text:
Assume a complete mapping n->S(n) where S(n) is supposed to cover all
subsets of N. Now consider the set P={k| k not in S(k)}. Clearly,
for every n only one of S(n) or P contains n as an element, and so P
is different from all S(n), proving the assumption wrong.
So now it is 4 lines. And one does not need more than that.
>> And that's what you call "with no justification that I can see".
>
> No, it is not -- what I called "with no justification that I can
> see" was something else:
>
> It was the assertion that no flaw having been found in a proof leads
> to a certainty that such a flaw cannot exist. I still see no
> justification for that.
Fine, so you think that a four-liner that has been out and tested for
hundreds of years by thousands of competent mathematicians provides no
justification for some statement.
Just what _would_ constitute justification in your book?
Jesse F. Hughes
> David Kastrup <d...@gnu.org> wrote:
>
>> Alec McKenzie <mcke...@despammed.com> writes:
>> > It has been known for a proof to be put forward, and fully accepted
>> > by the mathematical community, with a fatal flaw only spotted years
>> > later.
>>
>> In a concise 7 line proof? Bloody likely.
>
> I doubt it had seven lines, but I really don't know how many.
> Probably many more than seven.
It is easily formalized. It is a remarkably short and simple proof
and does not require any large body of theorems to reach its
conclusion. (It does require unpacking of the definitions of
"function" and "onto", of course, and this is tedious but not
difficult.)
I don't know what version of the proof David has in mind, but I'm sure
that a formal version is longer than seven lines. Nonetheless,
precisely because it is formal, that version can be easily checked for
correctness.
>> And that's what you call "with no justification that I can see".
>
> No, it is not -- what I called "with no justification that I can
> see" was something else:
>
> It was the assertion that no flaw having been found in a proof
> leads to a certainty that such a flaw cannot exist. I still see
> no justification for that.
What is your standard of justification? What more should a person do
than provide the proof?
Seems like you want a proof that the proof is correct in addition to
the proof itself. Of course, we would want further proof that this
verification is correct. And then...
In any case, none of this applies much to Cantor's remarkably simple
argument. Just work through it in first order logic. Nothing to it!
Are you waiting for someone else to do this work for you?
Well, no problem. I understand it's been done in Isabelle, for
instance. See <http://arxiv.org/pdf/cs.LO/9311103>. But I'm also
sure that there are other formalizations lying around.
--
Jesse F. Hughes
"We need to counter the shockwave of the evildoer by having individual
rate cuts accelerated and by thinking about tax rebates."
-- George W. Bush, Oct. 4, 2001
Jesse F. Hughes
> Infinite sets are equivalent.
>
> In one sense that's constructively illustrable via induction, with
> well-ordering and transfer. That's basically a three line proof that
> infinite sets are equivalent.
Great!
What are the three lines?
[...]
> V = L anyways. Appealing to constructibility, or finitudinous, finity,
> does not avoid variously these consequences, with infinity, and the
> infinite universe is the infinite constructible universe.
Illuminating!
> Obviously I suggest the null axiom theory. ZF is inconsistent.
Indeed! But what is the proof of inconsistency again?
--
Jesse F. Hughes
"My experience indicates that the people who post on this newsgroup
are about at the level of a 10 year old in the year 2060."
-- More wisdom from James Harris, time traveler
Patrick
Not a 5-liner.
David Kastrup
> Le 19/07/05 16:20, dans 85zmsic...@lola.goethe.zz, « David Kastrup »
> <d...@gnu.org> a écrit :
>
>> Alec McKenzie <mcke...@despammed.com> writes:
>>
>>> David Kastrup <d...@gnu.org> wrote:
>>>
>>>> Alec McKenzie <mcke...@despammed.com> writes:
>>>>> It has been known for a proof to be put forward, and fully accepted
>>>>> by the mathematical community, with a fatal flaw only spotted years
>>>>> later.
>>>>
>>>> In a concise 7 line proof? Bloody likely.
>>>
>>> I doubt it had seven lines, but I really don't know how many.
>>> Probably many more than seven.
>>
>> It was seven lines in my posting. You probably skipped over it. It
>> is a really simple and concise proof. Here it is again, for the
>> reading impaired, this time with a bit less text:
>>
>> Assume a complete mapping n->S(n) where S(n) is supposed to cover all
>> subsets of N. Now consider the set P={k| k not in S(k)}. Clearly,
>> for every n only one of S(n) or P contains n as an element, and so P
>> is different from all S(n), proving the assumption wrong.
>
> Not sure what you call a complete mapping. If it's a surjective
> application, then you only proved that you cannot find one. Big
> deal. Hint: it has been known since the early days of set theory.
Well, tell that to Alec.
>> So now it is 4 lines. And one does not need more than that.
>>
>>>> And that's what you call "with no justification that I can see".
>>>
>>> No, it is not -- what I called "with no justification that I can
>>> see" was something else:
>>>
>>> It was the assertion that no flaw having been found in a proof leads
>>> to a certainty that such a flaw cannot exist. I still see no
>>> justification for that.
>>
>> Fine, so you think that a four-liner that has been out and tested for
>> hundreds of years by thousands of competent mathematicians provides no
>> justification for some statement.
>
> Depends on which statement. If you want to prove that set theory is
> not coherent, you failed.
Well, tell that to Alec.
Perhaps you need to reread the thread and find out who is writing what?
Kevin Delaney
david petry wrote:
> I'm in the process of writing an article about
> objections to Cantor's Theory, which I plan to contribute
> to the Wikipedia. I would be interested in having
>
It seems to me that gap between "Cantorians" and "Anti-Cantorians" is
pretty much the same gap between classical and modern thought as
existed in most subjects in the 20th century. As I recall, Wallace
described transfinite theory as the final nail in Aristotle's coffin.
Transfinite theory was the primary reason for removing the study of
logic, grammar and arithematic in American public education and
replacing it with new math.
There are some who think the world was on a better thread with the
classical tradition.
> The pure mathematicians tend to view mathematics as an art
> form. They seek to create beautiful theories,
Many of the anti-Cantorians are such because they find the theory to be
ugly. Poincare described it as a disease. Brouwer had similar
invective.
I think it isn't quite accurate to say that the people who dislike the
theory are completely devoid of aesthetics. A better description is
that they are petty bourgeoisie. Cantorians, of course, are avante
garde revolutionaries.
As pointed out in a different post. The people who are opposed to a
theory are generally a lot more diverse than those who support it. For
example, Brouwer was opposed to the law of excluded middle. I suspect
that others are hoping to pull off a stunt like Russell. Russell's
early work on the reflexive paradox could be considered
"anti-Cantorian". Russell's two step catapulted him to the top of the
intellectual community and led to the decline of Frege. I suspect that
many people toy with anti-Cantorian thoughts because they hope to
repeat the trick.
Keith A. Lewis
>Keith A. Lewis wrote:
>> Stephen Montgomery-Smith <ste...@math.missouri.edu> writes in article <yUWCe.163211$x96.36939@attbi_s72> dated Mon, 18 Jul 2005 23:26:22 GMT:
>
>
>>>I hope that you are not yourself an anti-Cantorian whom I have
>>>inadvertantly offended, or if you are I would certainly be interested in
>>>hearing a non-crackpot approach against Cantor's arguments.
>>
>>
>> As an "undecided" myself, I have three:
>>
>> 1. If a number is indescribeable and unrepresentable, it is irrelevant.
>> The set of relevant numbers has cardinality equal to the naturals.
>
>Actually, what is a representable number? Are all the numbers between 1
>and googolplex representable? In principle, yes? In practice,
>certainly no.
>
>And if I do accept your paradigm, I find it hard to ascribe any real
>sense to your second statement. What does cardinality equal to the
>naturals mean?
We communicate our knowledge in discrete symbols. We have devised ways of
discussing irrationals using those symbols, but only a countable number of
them. Cantor proved that there are a whole lot more reals in between the
ones we know about, but ... is there any point in talking about them if we
can't even represent one?
>> 2. No continuum has been discovered in physics -- everything seems to
>> change in finite units called quanta. That's the real world.
>
>Yes, but those quanta seem to come in continuum like amounts - for
>example, is the ratio of the different energy levels of an electron
>necessarily a rational number? (I actually have no idea what the answer
>is.)
Based on my college chemistry of 20 years ago, the difference in energy
levels is always a multiple of a quanta, a constant discovered by
Heisenberg. Or was it just theorized by Heisenberg and discovered by
somebody later?
Somebody else said there were continua in quantum physics, maybe that's
true. Also time may be a continuum.
Wade said that whether something exists in physics is irrelevant to whether
it can exist in mathematics. That's true.
>> 3. The Dedekind cut paradox. Rationals on [0,1] are totally ordered and
>> countable, yet there are an uncountable number of cuts based on this total
>> ordering. With any finite set of cardinality n, there are n+1 such cuts.
>> But N u {-1} is countable, so the set of Dedekind cuts should be.
>
>Similarly all finite dimensional matrices have discrete spectra. But
>infinite dimensional operators can have spectra that are even of the
>second Baire category. There is a real problem with extrapolating
>finitary ideas to the infinite. "Taking the limit" in a naive manner
>doesn't always work.
It still bugs me!
But going back to crackpots vs. regular people who doubt Cantor -- I have my
doubts, and for now they're going to stay. This doesn't mean I can't
discuss cardinality but I'm certainly not going to spend a huge amount of
time debating the issue. Thanks to all who replied.
>My feelings are that there are genuine philosophical problems with the
>current approach of modern mathematics. But my sense is that our
>current ideas about philosophy are so primitive, that discussions that
>try to resolve these issues can never rise above scholastic arguments
>about how many angels can fit on a pinhead. Right now the current
>approach to mathematics is extremely effective and useful. My
>suggestion is that we wait another five hundred years before trying to
>resolve these issues.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
Alec McKenzie
> Ok, here's another question. Suppose that we want to
> prove that A implies B. Suppose that we have an
> completely flawless proof that A implies C, and
> a completely flawless proof that C implies B.
> Does the union of those two proofs constitute
> a flawless proof that A implies B?
Yes, I would say it does.
> I imagine you'll say yes to that as well. But
> the proof of the theorem in question really
> does involve nothing more than statements
> which are true by definition and statements
> which follow from previous statements by
> "if A implies C and C implies B then A implies
> B" arguments.
I would expect that to be the case for most direct proofs, if
not all.
In the case of the proof of the theorem in question, we do not
already know for a fact that the conclusion is true; neither do
we know that it is false. If we did already know it was true
there would be little point in trying to find a flaw, regardless
of whether one might exist (there could be a flaw in the proof,
even if the conclusion is correct). But if we knew it to be
false, the flaw would have to be there even if we cannot find it.
There are also proofs where we do know for a fact the conclusion
is false, even though there is no apparent flaw. The paradox of
the unexpected examination is an example of this, and I think
that if the conclusion in that case had been one, like Cantor,
where we had no way of knowing (apart from the proof) whether
the conclusion were true or false, the validity of the proof
would be almost universally accepted, and just as vigorously
defended.
Equally, if the conclusion of Cantor's proof were known for a
fact to be false, it would be known as Cantor's paradox.
My own feeling is that there exists the possibility, however
slight, that Cantor's conclusion is an obscure manifestation of
a paradox.
--
Alec McKenzie
mcke...@despammed.com
Alec McKenzie
> Alec McKenzie <mcke...@despammed.com> writes:
>
> > David Kastrup <d...@gnu.org> wrote:
> >
> >> Alec McKenzie <mcke...@despammed.com> writes:
> >> > It has been known for a proof to be put forward, and fully accepted
> >> > by the mathematical community, with a fatal flaw only spotted years
> >> > later.
> >>
> >> In a concise 7 line proof? Bloody likely.
> >
> > I doubt it had seven lines, but I really don't know how many.
> > Probably many more than seven.
>
> It was seven lines in my posting. You probably skipped over it.
I see you have misunderstood what I said. You seemed to be
denying that the accepted proof I mentioned (that turned out to
have a flaw) was only seven lines, and I was merely saying that
I didn't know how many lines it had.
--
Alec McKenzie
mcke...@despammed.com
Michael Stemper
>Cantor's Theory, if taken seriously, would lead us to believe
>that while the collection of all objects in the world of
>computation is a countable set, and while the collection of all
>identifiable abstractions derived from the world of computation
>is a countable set,
To be consistent with the philosophy that only things that can physically
exist are meaningful, you shouldn't say "countable" here, you should say
"finite".
After all, there are only about 10^78 atoms in the universe, which puts
a very definite (and finite) cap on how many computers there can be. In
addition to that, each of those computers can only be in a finite number
of possible states. Therefore, there's only (by this philosophy) a finite
number of "identifiable abstractions derived from the world of computation",
rather than a countable number, as you stated.
--
Michael F. Stemper
#include <Standard_Disclaimer>
A bad day sailing is better than a good day at the office.
The World Wide Wade
David C. Ullrich <ull...@math.okstate.edu> wrote:
> I once had a person tell me the following, with a straight face:
>
> (*) "You can't say for sure there's no such thing as a square
> circle! I mean just because they haven't found one yet doesn't
> mean they won't discover one tomorrow."
>
> Please choose one of the following replies:
>
> (i) No, (*) is nonsense. If it's square then _by definition_
> it's not a circle. So they will _never_ find a square circle.
>
> (ii) Hmm, good point.
>
> You really should choose one of (i) or (ii), so people know
> how to reply to your post. The point: If you say (ii) then
> we know that there's no point worrying about anything you
> say. Otoh if you say (i) then there's hope - you agree that
> we're _certain_ they will never find a square circle, now
> we just have to convince you that our assertions about
> enumerations of subsets of N are just as certain, for
> entirely similar (although slightly more complicated)
> reasons.
You need to modify (i) because the definition of a square nowhere
specifies a square is not a circle.
Jesse F. Hughes
> Transfinite theory was the primary reason for removing the study of
> logic, grammar and arithematic in American public education and
> replacing it with new math.
Fascinating. Got any citations for that?
Wow. Grammar was removed from public education due to Cantor's
theorem. Who'da thunk it?
> There are some who think the world was on a better thread with the
> classical tradition.
There are some who think that one shouldn't make things up and call it
an argument. But they're not as persuasive as you.
[...]
> As pointed out in a different post. The people who are opposed to a
> theory are generally a lot more diverse than those who support
> it. For example, Brouwer was opposed to the law of excluded
> middle. I suspect that others are hoping to pull off a stunt like
> Russell. Russell's early work on the reflexive paradox could be
> considered "anti-Cantorian". Russell's two step catapulted him to
> the top of the intellectual community and led to the decline of
> Frege. I suspect that many people toy with anti-Cantorian thoughts
> because they hope to repeat the trick.
Decline of Frege? Gosh.
Of course, Russell continued to be indebted to Frege and Frege
continues to be a pillar of philosophy of mathematics. Well, not
quite. Frege's reputation grew considerably after his death (and long
after Russell's paradox was discovered), if I understand correctly.
--
Jesse F. Hughes
"Contrariwise," continued Tweedledee, "if it was so, it might be, and
if it were so, it would be; but as it isn't, it ain't. That's logic!"
-- Lewis Carroll
Kevin Delaney
on teaching grammar, logic, arithemetic and rhetoric.
The modern era felt that these were all artificial edifaces. There was
a concerted effort to pull these subjects out of the schools. They were
there. They were not there when I went to school.
You can verify this if you look at the standard school curriculum in
say 1900 and compare it to the curriculum in, say, the 1970s. I used to
do stupid things like look up the different books that were taught at
different times to try and see how different eras would see the world.
Transfinite theory is not the only manifestation of modern thinking.
So, I am not saying that transfinite theory alone was the cause for
this transformation. The basic idea was that traditional logic, grammar
and arithematic were part of this horrible weight keeping people down,
and that we would transcend to a higher level of existence.
Citing all the places where people attacked classical education would
take several years. It is pretty much a fact that in the modern times
there was one curriculum replaced by another curriculum. In some cases
it was a postive thing, curriculums always need improvement and
adjustment. The problem with a wholesale replacement is that you end up
losing the promising threads of the previous curriculum.
Tony Orlow
> "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
>
> > Can anti-Cantorians identify correctly a flaw in the proof that there
> > exists no enumeration of the subsets of the natural numbers?
>
> In my view the answer to that question a definite "No, they
> can't".
>
> However, the fact that no flaw has yet been correctly identified
> that is just what pro-Cantorians appear to be asserting, with no
>
>
number where the first bit denotes membership of the first element, the second
bit denotes membership of the second element, etc? The only objection to this
bijection between the natural numbers and the subsets of the natural numbers is
the nonsensical insistence that every natural number in the infinite set is
finite, which is mathematically impossible, given the fact that each additional
element requires a constant incrementation of the entire range of values in the
set. You may all pat each other on the back and dismiss "anti-Cantorians" as
cranks, but that is only with a diligent lack of attention to every other area
of mathematics and logic.
--
Smiles,
Tony
Stephen Montgomery-Smith
> "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
>
>
>>Can anti-Cantorians identify correctly a flaw in the proof that there
>>exists no enumeration of the subsets of the natural numbers?
>
>
> In my view the answer to that question a definite "No, they
> can't".
>
> However, the fact that no flaw has yet been correctly identified
> does not lead to a certainty that such a flaw cannot exist. Yet
> that is just what pro-Cantorians appear to be asserting, with no
> justification that I can see.
As best I can see from your other posts, you are making one of two points:
1. There is such an enumeration, because set thoery is inconsistent.
Yes, I cannot be sure that cannot happen, but it would not invalidate
the proof (because the theorem would be both true and not true
simultaneously). If that is your problem, I think your issue is with
proof by contradiction, not with Cantor's argument us such.
2. There really is a flaw in the proof, but mathematicians have somehow
simply not seen it. While one cannot totally discount this possibility,
the chances of this being the case is so extraordinarily small that for
all practical purposes it is just not the case. We are talking
probabilities like that of a Monkey sitting at a typewriter and dashing
off a Shakespeare play. In principle, yes it can happen, in reality,
you should worry more about UFO's abducting you.
Stephen
Tony Orlow
> Alec McKenzie <mcke...@despammed.com> writes:
>
> > "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
> >
> >> Can anti-Cantorians identify correctly a flaw in the proof that there
> >> exists no enumeration of the subsets of the natural numbers?
> >
> > In my view the answer to that question a definite "No, they
> > can't".
> >
> > However, the fact that no flaw has yet been correctly identified
> > does not lead to a certainty that such a flaw cannot exist. Yet
> > that is just what pro-Cantorians appear to be asserting, with no
> > justification that I can see.
>
>
> The proof of Cantor's theorem is easily formalized. It's remarkably
> short and simple and every step can be verified as correct.
>
> It is perfectly reasonable to assert that no such flaw exists (given
> the axioms used in the proof). Indeed, why would anyone entertain any
> doubts when he can confirm the correctness of each and every step of
> the proof?
>
>
In all actuality, the flaws in various proofs and assumptions in set theory
have been directly addressed, and ignored by the mainstream thinkers here.
Now, I am not familiar, I think, with the proof concerning subsets of the
natural numbers. Certainly a power set is a larger set than the set it's
derived from, but that is no proof that it cannot be enumerated. Is this the
same as the proof concerning the "uncountability" of the reals?
--
Smiles,
Tony
Tony Orlow
> Alec McKenzie <mcke...@despammed.com> writes:
>
> > "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
> >
> >> Can anti-Cantorians identify correctly a flaw in the proof that
> >> there exists no enumeration of the subsets of the natural numbers?
> >
> > In my view the answer to that question a definite "No, they can't".
> >
> > However, the fact that no flaw has yet been correctly identified
> > does not lead to a certainty that such a flaw cannot exist.
>
>
> Then consider the set of all naturals that are not member of the
> subset which they map to.
>
> The membership of each natural can be clearly established from the
> mapping, and it is clearly different from the membership of the
> mapping indicated by the natural. So the assumption of a complete
> mapping was invalid.
>
> > justification that I can see.
>
> need apart from a clear 7-line proof? It simply does not get better
> than that.
>
>
the natural number corresponding to the subset to be a member of that subset.
if this rests on the diagonal proof, there is a very clear flaw in that proof
which you folks simply dismiss as irrelvant, but which is fatal. Still,
discussing these things with Cantorians is like trying to discuss evolution
with an evangelical christian.
--
Smiles,
Tony
David Kastrup
> Alec McKenzie said:
>> "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
>>
>> > Can anti-Cantorians identify correctly a flaw in the proof that there
>> > exists no enumeration of the subsets of the natural numbers?
>>
>> In my view the answer to that question a definite "No, they
>> can't".
>>
>> However, the fact that no flaw has yet been correctly identified
>> does not lead to a certainty that such a flaw cannot exist. Yet
>> that is just what pro-Cantorians appear to be asserting, with no
>> justification that I can see.
>>
> Even though every subset of the natural numbers can be represented
> by a binary number where the first bit denotes membership of the
> first element, the second bit denotes membership of the second
> element, etc?
Well, what number will then represent the set of numbers dividable by
three?
> The only objection to this bijection between the natural numbers and
> the subsets of the natural numbers is the nonsensical insistence
> that every natural number in the infinite set is finite, which is
> mathematically impossible, given the fact that each additional
> element requires a constant incrementation of the entire range of
> values in the set.
You are babbling. Anyway, let's assume just for kicks that infinite
numbers are part of the natural numbers, and lets take your numbering
scheme.
Let's take the number representing the set of numbers dividable by
three. Is this number dividable by three?
Chan-Ho Suh
McKenzie <mcke...@despammed.com> wrote:
> I quite agree that it does not get better than that, but I think
> one must allow some room for doubt, however small, for any
> proof. Otherwise one is proclaiming infallibility.
>
No one has proclaimed infallibility. People have espoused the view
that "it does not get better than that".
No one is denying that humans are fallible. Your memory of your entire
life is distorted and possibly even a delusion. You cannot know for
certain. If that was your point, it's such an obvious and silly point
as to not bear making a big deal out of. And it's certainly irrelevant
to bring it into this discussion.
> It has been known for a proof to be put forward, and fully
> accepted by the mathematical community, with a fatal flaw only
> spotted years later.
I doubt this. "Fully accepted" means that the community either was not
paying attention or didn't care enough to check themselves, etc. I
know of no proof in the modern literature that was verified correct by
a large number of experts and with a flaw only found years later. And
there certainly have been no results commonly found in modern undergrad
textbooks, which have been verified by virtually every research
mathematician and found to have a flaw only years later.
It's rather irrelevant to this discussion to bring up any kind of proof
that only a dozen people have verified. That's a totally different
situation.
David Kastrup
> Jesse F. Hughes said:
>> Alec McKenzie <mcke...@despammed.com> writes:
>>
>> > "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
>> >
>> >> Can anti-Cantorians identify correctly a flaw in the proof that there
>> >> exists no enumeration of the subsets of the natural numbers?
>> >
>> > In my view the answer to that question a definite "No, they
>> > can't".
>> >
>> > However, the fact that no flaw has yet been correctly identified
>> > does not lead to a certainty that such a flaw cannot exist. Yet
>> > that is just what pro-Cantorians appear to be asserting, with no
>> > justification that I can see.
>>
>> Huh?
>>
>> The proof of Cantor's theorem is easily formalized. It's remarkably
>> short and simple and every step can be verified as correct.
>
> In all actuality, the flaws in various proofs and assumptions in set
> theory have been directly addressed, and ignored by the mainstream
> thinkers here.
Guffaw.
> Now, I am not familiar, I think, with the proof concerning subsets
> of the natural numbers. Certainly a power set is a larger set than
> the set it's derived from, but that is no proof that it cannot be
> enumerated.
Uh, not?
> Is this the same as the proof concerning the "uncountability" of the
> reals?
It's pretty similar.
Assume a set X can be put into complete bijection with its powerset
P(X) such that we have a mapping x->f(x) where x is an element from X
and f(x) is an element from P(X). Now consider
Q = {x in X|x not in f(x)}. Clearly, for all x in X we have
Q unequal to f(x), since x is a member of exactly one of f(x) and Q.
So Q is missing from the bijection.
Robert Low
> Certainly a power set is a larger set than the set it's
> derived from, but that is no proof that it cannot be enumerated. Is this the
> same as the proof concerning the "uncountability" of the reals?
So, you accept that the power set of the naturals is bigger
than the set of naturals, but also think that the power
set of the naturals can perhaps be enumerated.
What do you mean by 'bigger than' in this context?
Robert Low
> David Kastrup said:
> I still do not get this. You have a set of naturals {0,1,2,3...}, and a set of
> binary numbers {0,1,10,11,100,101,110,111,....}. Surely there is a bijection
> between these two sets.
Of course. But that isn't the issue. The issue is the existence
of a bijection between either of these sets and the power set
of the set of naturals. So, which binary number does your
procedure associate with the set of *all* natural numbers
divisible by 3?
Chan-Ho Suh
<je...@phiwumbda.org> wrote:
> David C. Ullrich <ull...@math.okstate.edu> writes:
>
> > Ok, here's another question. Suppose that we want to
> > prove that A implies B. Suppose that we have an
> > completely flawless proof that A implies C, and
> > a completely flawless proof that C implies B.
> > Does the union of those two proofs constitute
> > a flawless proof that A implies B?
>
> Am I the only one suffering from flashbacks of Achilles and the
> Tortoise?
>
>
No. :-)
David Kastrup
> David Kastrup said:
>> Alec McKenzie <mcke...@despammed.com> writes:
>>
>> > "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
>> >
>> >> Can anti-Cantorians identify correctly a flaw in the proof that
>> >> there exists no enumeration of the subsets of the natural numbers?
>> >
>> > In my view the answer to that question a definite "No, they can't".
>> >
>> > However, the fact that no flaw has yet been correctly identified
>> > does not lead to a certainty that such a flaw cannot exist.
>>
>> Uh, what? There is nothing fuzzy about the proof.
>>
>> Suppose that a mapping of naturals to the subsets of naturals exists.
>> Then consider the set of all naturals that are not member of the
>> subset which they map to.
>>
>> The membership of each natural can be clearly established from the
>> mapping, and it is clearly different from the membership of the
>> mapping indicated by the natural. So the assumption of a complete
>> mapping was invalid.
>>
>> > Yet that is just what pro-Cantorians appear to be asserting, with no
>> > justification that I can see.
>>
>> Uh, where is there any room for doubt? What more justification do you
>> need apart from a clear 7-line proof? It simply does not get better
>> than that.
> Is the above your 7-line proof? it makes no sense.
If you don't get it.
> There is no reason to expect the natural number corresponding to the
> subset to be a member of that subset.
There is no such expectation. The only expectation is that _every_
natural number is _either_ a member of its corresponding subset, or
not. _Depending_ on that, the constructed subset will either _not_ or
_do_ contain the number, respectively. This constructed subset then
does not correspond to _any_ natural number.
> if this rests on the diagonal proof,
Rather the other way round. It is more basic than the diagonal proof.
Tony Orlow
> >> > by the mathematical community, with a fatal flaw only spotted years
> >> > later.
> >>
> >
> > I doubt it had seven lines, but I really don't know how many.
> > Probably many more than seven.
>
> is a really simple and concise proof. Here it is again, for the
> reading impaired, this time with a bit less text:
>
> Assume a complete mapping n->S(n) where S(n) is supposed to cover all
> subsets of N. Now consider the set P={k| k not in S(k)}. Clearly,
> for every n only one of S(n) or P contains n as an element, and so P
> is different from all S(n), proving the assumption wrong.
binary numbers {0,1,10,11,100,101,110,111,....}. Surely there is a bijection
numbers (with implied leading zeroes) as being a map of each subset, where each
successive bit represents membership in thesubset by each successive natural
number? So, 0 represents the null set, 1 represents the set including the first
element, 10 includes just the second, 11 includes the second and third, etc..
You seem to be complaining that the fourth subset, with a numerical value of 3,
only includes the first and second elements?
I mean, your statement that begins with "Clearly" is not at all clear to me.
Any binary number which has a 1 in the rightmost digit contains the first
element, so the first element is a member of an infinite number of such
subsets. What is the assumption which is supposedly wrong?
>
> So now it is 4 lines. And one does not need more than that.
>
> >> And that's what you call "with no justification that I can see".
> >
> > No, it is not -- what I called "with no justification that I can
> > see" was something else:
> >
> > It was the assertion that no flaw having been found in a proof leads
> > to a certainty that such a flaw cannot exist. I still see no
> > justification for that.
>
> Fine, so you think that a four-liner that has been out and tested for
> hundreds of years by thousands of competent mathematicians provides no
> justification for some statement.
Just over 100 years, and we've learned a lot since then. This isn't religion.
>
> Just what _would_ constitute justification in your book?
>
>
--
Smiles,
Tony
Chan-Ho Suh
McKenzie <mcke...@despammed.com> wrote:
Aha, so you're saying you had a particular proof in mind? Well, let's
hear which one it is.
Chan-Ho Suh
<tor...@sm.luth.se> wrote:
> Jesse F. Hughes wrote:
>
> > Really, Torkel, whether you are impressed by Wikipedia or not, you
> > shouldn't encourage folks to deface the site.
>
> Well, rants are unlikely to survive long in Wikipedia.
You don't have very much experience with Wikipedia. Rants and
dicussions with the ranters can survive for a long time. Generally
this is because those dealing with the ranters are idealistic and try
to discuss things reasonably with them...oh wait, that sounds familiar
for some reason.
> So if somebody asks for feedback on a proposed rant, surely the
> simple and reasonable course is simply to encourage him to
> post it.
>
I hear Wikipedia editors don't need very much sleep.
Chan-Ho Suh
McKenzie <mcke...@despammed.com> wrote:
> David Kastrup <d...@gnu.org> wrote:
>
> > Alec McKenzie <mcke...@despammed.com> writes:
> > > It has been known for a proof to be put forward, and fully accepted
> > > by the mathematical community, with a fatal flaw only spotted years
> > > later.
> >
> > In a concise 7 line proof? Bloody likely.
>
> I doubt it had seven lines, but I really don't know how many.
> Probably many more than seven.
>
Not really. But if you are trying to assert that this is a complicated
proof, significantly more complicated than a proof of say, the
irrationality of two, then you are mistaken.
> > And that's what you call "with no justification that I can see".
>
> No, it is not -- what I called "with no justification that I can
> see" was something else:
>
> It was the assertion that no flaw having been found in a proof
> leads to a certainty that such a flaw cannot exist. I still see
> no justification for that.
Nobody asserted this. People have asserted that a particular given
proof has no flaw. Of course, nothing is absolutely certain. Just as
you cannot be absolutely certain you exist or that your memory of what
you ate for lunch is correct. But if your point is the rather trivial
point that nothing is absolutely certain, then there was no need to
make it.
David Kastrup
> David Kastrup said:
>> Alec McKenzie <mcke...@despammed.com> writes:
>>
>> > David Kastrup <d...@gnu.org> wrote:
>> >
>> >> Alec McKenzie <mcke...@despammed.com> writes:
>> >> > It has been known for a proof to be put forward, and fully accepted
>> >> > by the mathematical community, with a fatal flaw only spotted years
>> >> > later.
>> >>
>> >> In a concise 7 line proof? Bloody likely.
>> >
>> > I doubt it had seven lines, but I really don't know how many.
>> > Probably many more than seven.
>>
>> It was seven lines in my posting. You probably skipped over it. It
>> is a really simple and concise proof. Here it is again, for the
>> reading impaired, this time with a bit less text:
>>
>> Assume a complete mapping n->S(n) where S(n) is supposed to cover all
>> subsets of N. Now consider the set P={k| k not in S(k)}. Clearly,
>> for every n only one of S(n) or P contains n as an element, and so P
>> is different from all S(n), proving the assumption wrong.
> I still do not get this. You have a set of naturals {0,1,2,3...},
> and a set of binary numbers {0,1,10,11,100,101,110,111,....}. Surely
> there is a bijection between these two sets.
Fine.
> So, what is the problem if one interprets the binary numbers (with
> implied leading zeroes) as being a map of each subset, where each
> successive bit represents membership in thesubset by each successive
> natural number?
Ok, so let's construct P. It is actually easy enough, since n<2^n,
and so P=N. So what number corresponds to N itself in your mapping?
Tony Orlow
> Tony Orlow (aeo6) <ae...@cornell.edu> writes:
>
> > Alec McKenzie said:
> >> "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
> >>
> >> > Can anti-Cantorians identify correctly a flaw in the proof that there
> >> > exists no enumeration of the subsets of the natural numbers?
> >>
> >> In my view the answer to that question a definite "No, they
> >> can't".
> >>
> >> However, the fact that no flaw has yet been correctly identified
> >> does not lead to a certainty that such a flaw cannot exist. Yet
> >> that is just what pro-Cantorians appear to be asserting, with no
> >> justification that I can see.
> >>
> > Even though every subset of the natural numbers can be represented
> > by a binary number where the first bit denotes membership of the
> > first element, the second bit denotes membership of the second
> > element, etc?
>
> Well, what number will then represent the set of numbers dividable by
> three?
Of course, you will argue that this infinite value is not a natural number,
since all naturals are finite, but that is clearly incorrect, as it is
impossible to have an infinite set of values all differing by a constant finite
amount from their neighbors, and not have an overall infinite difference
between some pair of them, indicating that at least one of them is infinite.
>
> > The only objection to this bijection between the natural numbers and
> > the subsets of the natural numbers is the nonsensical insistence
> > that every natural number in the infinite set is finite, which is
> > mathematically impossible, given the fact that each additional
> > element requires a constant incrementation of the entire range of
> > values in the set.
>
> You are babbling. Anyway, let's assume just for kicks that infinite
> numbers are part of the natural numbers, and lets take your numbering
> scheme.
nonsense. Please try to be civil. I notice that the Cantorians routinely accuse
any opponent of babling, spewing nonsense being a crackpot or crank, or other
personal insults, rather than replying with any form of logic. It's a
convenient way of changing the subject.
>
> Let's take the number representing the set of numbers dividable by
> three. Is this number dividable by three?
Why does it have to be? There is no requirement that the number that represents
a subset of numbers be a member of that subset of numbers, any more than it is
a requirement that the natural number corresponding to a given rational number
be either the numerator or denominator of that rational number, in Cantor's
diagonal proof of the countability of the rationals. Does the rational number
that is 10th in Cantor's enumeration need to have a 10 in it? No. That argument
is so pointless, it can't possibly be accepted as any proof of anything. Is it
really that bad? Have we sunk so low, that we prove things based on ideas that
don't even have a semblance of sense? I concur with Poincare and Bouwer.
>
>
--
Smiles,
Tony
Chris Menzel
> ...
> to expect the natural number corresponding to the subset to be a
> member of that subset. if this rests on the diagonal proof, there is
> a very clear flaw in that proof which you folks simply dismiss as
> irrelvant, but which is fatal.
There is a simple, demonstrably valid proof of Cantor's Theorem in ZF
set theory. So you must think the proof is unsound. Which axiom of ZF
do you believe to be false?
Chris Menzel
Shmuel (Seymour J.) Metz
at 12:27 AM, kle...@OMEGA.MITRE.ORG (Keith A. Lewis) said:
>2. No continuum has been discovered in physics -- everything seems
>to change in finite units called quanta. That's the real world.
No classical particles have been discovered in Physics -- everything
seems to manifest wave-like behavior. That's the real world. Solar
spectra may have some absorption and emission lines, but to within the
accuracy of our instruments they are continuous.
>3. The Dedekind cut paradox.
There is no paradox there, only faulty intuition.
>Rationals on [0,1] are totally ordered and countable, yet there are
>an uncountable number of cuts based on this total ordering.
Correct so far.
>With any finite set of cardinality n, there are n+1 such cuts.
Correct so far. Many things are true of finite sets that are not true
in general.
>But N u {-1} is countable, so the set of Dedekind cuts should be.
No it shouldn't. You're trying to carry over into the infinite case
something that is only true for the finite c ase.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
David Kastrup
You have not shown such a thing, and of course it would be
inconsistent with the Peano axioms defining the naturals.
>> Let's take the number representing the set of numbers dividable by
>> three. Is this number dividable by three?
>
> Why does it have to be?
It does not have to be. But if it is a natural number, it either is
dividable by three, or it isn't. You claim that it is a natural
number. So what is it? Is it dividable by three, or isn't it?
It must be one, mustn't it?
Tony Orlow
> Tony Orlow (aeo6) <ae...@cornell.edu> writes:
>
> > Jesse F. Hughes said:
> >> Alec McKenzie <mcke...@despammed.com> writes:
> >>
> >> > "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
> >> >
> >> >> Can anti-Cantorians identify correctly a flaw in the proof that there
> >> >> exists no enumeration of the subsets of the natural numbers?
> >> >
> >> > In my view the answer to that question a definite "No, they
> >> > can't".
> >> >
> >> > However, the fact that no flaw has yet been correctly identified
> >> > does not lead to a certainty that such a flaw cannot exist. Yet
> >> > that is just what pro-Cantorians appear to be asserting, with no
> >> > justification that I can see.
> >>
> >> Huh?
> >>
> >> The proof of Cantor's theorem is easily formalized. It's remarkably
> >> short and simple and every step can be verified as correct.
> >
> > In all actuality, the flaws in various proofs and assumptions in set
> > theory have been directly addressed, and ignored by the mainstream
> > thinkers here.
>
> Guffaw.
>
> > Now, I am not familiar, I think, with the proof concerning subsets
> > of the natural numbers. Certainly a power set is a larger set than
> > the set it's derived from, but that is no proof that it cannot be
> > enumerated.
>
> Uh, not?
that is a leap and an assumption.
>
> > Is this the same as the proof concerning the "uncountability" of the
> > reals?
>
> It's pretty similar.
>
> Assume a set X can be put into complete bijection with its powerset
> P(X) such that we have a mapping x->f(x) where x is an element from X
> and f(x) is an element from P(X). Now consider
> Q = {x in X|x not in f(x)}. Clearly, for all x in X we have
> Q unequal to f(x), since x is a member of exactly one of f(x) and Q.
> So Q is missing from the bijection.
>
>
just try to be clear, without hand-waving.
There is no requirement that subset number x include x as a member, despite the
fact that x is a member of an infinite number of subsets. Subset number 3,
represented as ...00011, includes only the first and second elements. What does
that prove? That 3 is not a element of subset 4 (...000100), which consists
ONLY of 3? I can't even imagine why this is thought to prove anything.
--
Smiles,
Tony
Tony Orlow
> _Depending_ on that, the constructed subset will either _not_ or
> _do_ contain the number, respectively.
> This constructed subset then
> does not correspond to _any_ natural number.
number denoted by the binary string constructed from the right to the left,
where each successive bit is a 1 if the successive natural is a member, and 0
if it is not. Any subset can be denoted as a string of bits, and any string of
bits can denote a natural number. The only reason to reject this bijection is
if one clings to the idea that all natural numbers are finite, which is
impossible.
>
> > if this rests on the diagonal proof,
>
> Rather the other way round. It is more basic than the diagonal proof.
making. It seems like you are assuming subset number X must contain X as a
member. If so, how do you justify this assumption?
>
>
--
Smiles,
Tony
David Kastrup
> David Kastrup said:
>> Tony Orlow (aeo6) <ae...@cornell.edu> writes:
>>
>> > Now, I am not familiar, I think, with the proof concerning
>> > subsets of the natural numbers. Certainly a power set is a larger
>> > set than the set it's derived from, but that is no proof that it
>> > cannot be enumerated.
>>
>> Uh, not?
> Yes, not. "Larger" is not a synonym for "uncountable" except in
> Cantorland, and that is a leap and an assumption.
"Larger" is a synonymon for "can't be surjected onto from" in set
theory. And "uncountable" is a synonymon for "larger than the set of
naturals". It is not a leap or an assumption, but simply a
definition.
>> > Is this the same as the proof concerning the "uncountability" of
>> > the reals?
>>
>> It's pretty similar.
> Figures.
>>
>> Assume a set X can be put into complete bijection with its powerset
>> P(X) such that we have a mapping x->f(x) where x is an element from X
>> and f(x) is an element from P(X). Now consider
>> Q = {x in X|x not in f(x)}. Clearly, for all x in X we have
>> Q unequal to f(x), since x is a member of exactly one of f(x) and Q.
>> So Q is missing from the bijection.
>>
>>
> Again with the "Clearly". You might want to refrain from using the
> word, and just try to be clear, without hand-waving.
>
> There is no requirement that subset number x include x as a member,
Quite so. But there is a requirement that subset number x _either_
include x as a member _or_ not include x as a member. Only one of
those two statements can be true. And then Q _either_ not includes x
as a member _or_ does include it, respectively.
You are free to choose your mapping as you want to. But once you have
chosen your mapping, each subset number x _either_ includes x as a
member _or_ it doesn't. Whether it does, can be chosen independently
for every x. But once you are through, for every particular x, x will
be in f(x), or it won't. And depending on that, x won't be in Q, or
it will.
Tony Orlow
naturals themselves, and then there are many more subsets. A power set of a set
with n elements has 2^n elements, which is always larger than n, for finite and
infinite sets.
The idea of uncountability as being equivalent to "larger than the set of
naturals" is unfounded. There is no reason to believe that larger sets cannot
be enumerated. the power set of the naturals can be enumerated and bijected
with the naturals, as I described in another post, as long as infinite natural
numbers are allowed.
--
Smiles,
Tony
Robert Low
> Robert Low said:
>
>>Tony Orlow (aeo6) wrote:
>>
>>> Certainly a power set is a larger set than the set it's
>>>derived from, but that is no proof that it cannot be enumerated. Is this the
>>>same as the proof concerning the "uncountability" of the reals?
>>
>>So, you accept that the power set of the naturals is bigger
>>than the set of naturals, but also think that the power
>>set of the naturals can perhaps be enumerated.
>>
>>What do you mean by 'bigger than' in this context?
>>
>
> I mean it has more elements.
And what does it mean to say 'the power set of N has more elements
than N', if it doesn't mean 'there is no surjection from
N to its power set'? Which is, of course, exactly what 'the
power set of N is uncountable' means...
> the power set of the naturals can be enumerated and bijected
> with the naturals, as I described in another post, as long as infinite natural
> numbers are allowed.
No, you can make a surjection from the set of (possibly infinite)
natural numbers to the power set of the set of *finite* integers.
If you also remember that you have to include those subsets
including your infinite integers, you find that you still
can't have a surjection from your set of (possibly infinite)
integers to the power set of that set. You're trying to have
your cake and eat it.
Tony Orlow
lack of time, but the diagonal proof suffers from the fatal flaw of assuming
that the diaginal traversal actually covers all the numbers in the list. Any
complete list of digital numbers of a given length, even a given infinite
length, is exponentially longer in members than wide in terms of the digits in
each member. Therefore, the diagonal traversal only shows that the anti-
diagonal does not exist in the first aleph_0 terms. Of course, the entire list
is presumed to be aleph_1 long, being a list of the reals, and the antidiagonal
simply exists on the list, below the line of diagonal traversal. Cantorians
seem to think infinity is simply infinity, even during the course of a proof
that that is not the case.
--
Smiles,
Tony
Tony Orlow
100100100...100100100. Notice, every third bit, from right to left, is a 1.
Those are the multiples of 3.
--
Smiles,
Tony
Tony Orlow
Infinite whole numbers are required for an infinite set of whole numbers.
--
Smiles,
Tony
Robert Low
> Robert Low said:
>>Of course. But that isn't the issue. The issue is the existence
>>of a bijection between either of these sets and the power set
>>of the set of naturals. So, which binary number does your
>>procedure associate with the set of *all* natural numbers
>>divisible by 3?
> Again, that infinite set is denoted by the infinite whole number
> 100100100...100100100. Notice, every third bit, from right to left, is a 1.
> Those are the multiples of 3.
So, you're saying that you can find a bijection between N and
its power set, as long as you get to say that N isn't what
everybody else thinks it is.
See elsewhere for the problem with this.
David Kastrup
> David Kastrup said:
>> Tony Orlow (aeo6) <ae...@cornell.edu> writes:
>>
>> > There is no reason to expect the natural number corresponding to
>> > the subset to be a member of that subset.
>>
>> There is no such expectation. The only expectation is that _every_
>> natural number is _either_ a member of its corresponding subset, or
>> not.
> Okay.
>> _Depending_ on that, the constructed subset will either _not_ or
>> _do_ contain the number, respectively.
> Redundant, but okay.
Sure it is redundant. But you should be the last person to complain
about your need to get this hammered bit by bit into your skull.
>> This constructed subset then does not correspond to _any_ natural
>> number.
> This is where it goes wrong.
Uh yes. That's the conclusion: that the assumption of the mapping
goes wrong, since it fails to cover the constructed subset.
> The consructed subset corresponds to the natural number denoted by
> the binary string constructed from the right to the left, where each
> successive bit is a 1 if the successive natural is a member, and 0
> if it is not.
You are assuming a particular mapping, the proof holds for _any_
mapping. So let's see where your mapping takes us. Since 2^n>n
always, the subset we get is simply N itself, at least as long as we
assume that N only contains finite numbers. However, that is an
assumption that you are not willing to make, so let is call this
particular subset of N by the name Q.
So what natural number corresponds to Q according to your logic? I'll
take a daring guess at your muddled thoughts and suppose 111...111 or
something like that. Is 111...111 itself a member of Q?
> It is so basic, I cannot even quite see what erroneous assumption
> you are making. It seems like you are assuming subset number X must
> contain X as a member. If so, how do you justify this assumption?
I don't. But I assume that for every X, subset number X must either
contain number X as a member, or doesn't. And if it does, subset Q
will not contain number X as a member, and if it doesn't, subset Q
will contain number X as a member, and so subset Q differs from every
subset X at the position of number X.
Alec McKenzie
> Alec McKenzie wrote:
> > "Stephen J. Herschkorn" <sjher...@netscape.net> wrote:
> >
> >
> >>Can anti-Cantorians identify correctly a flaw in the proof that there
> >>exists no enumeration of the subsets of the natural numbers?
> >
> >
> > In my view the answer to that question a definite "No, they
> > can't".
> >
> > However, the fact that no flaw has yet been correctly identified
> > does not lead to a certainty that such a flaw cannot exist. Yet
> > that is just what pro-Cantorians appear to be asserting, with no
> > justification that I can see.
>
> As best I can see from your other posts, you are making one of two points:
>
> 1. There is such an enumeration, because set thoery is inconsistent.
> Yes, I cannot be sure that cannot happen, but it would not invalidate
> the proof (because the theorem would be both true and not true
> simultaneously). If that is your problem, I think your issue is with
> proof by contradiction, not with Cantor's argument us such.
>
> 2. There really is a flaw in the proof, but mathematicians have somehow
> simply not seen it. While one cannot totally discount this possibility,
> the chances of this being the case is so extraordinarily small that for
> all practical purposes it is just not the case. We are talking
> probabilities like that of a Monkey sitting at a typewriter and dashing
> off a Shakespeare play. In principle, yes it can happen, in reality,
> you should worry more about UFO's abducting you.
I am not making either of those points, and I don't see why you
should think I am:
1. I have no reason to believe there is such an enumeration, and
I have never suggested that there is.
2. I have no reason to believe there really is a flaw in the
proof, and I have never suggested that there is.
The point I am trying to make is precisely the one I wrote:
"The fact that no flaw has yet been correctly identified does
not lead to a certainty that such a flaw cannot exist."
I still believe this to be true. Would you deny it?
--
Alec McKenzie
mcke...@despammed.com
David Kastrup
Is this number a member of the set of all natural numbers divisable by 3?