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george
science forum Guru Wannabe

Joined: 01 May 2005
Posts: 103

Posted: Thu Jul 13, 2006 5:43 pm    Post subject: Re: Question about Cantor's proof

Jiri Lebl wrote:
 Quote: Try going to a local bar and see how many people you convince with your proof.

Hmph -- try going to a bar and seeing how many people remember
what the concept of an infinite decimal is. Most people never
understood infinity right THE FIRST time.

 Quote: I had hard time explaining this proof to my dad and he's a phd (in chemistry).

 Quote: These are people who think that mathematical truth is guided by their intuition and their imagination. Those are precisely the two things you need to use to show that your attempt to count the reals has failed. No, I use formal logic. Intuition only gets me so far. If this was an intuitive result it would have been known far before Cantor.

What utter bullshit. It probably WAS known, actually.
You should see if Franz Fritsche will accept your
patronage for a search back through the 18th and 19th
century literature at large for examples of diagonal arguments.
The important point about the Cantorian argument/result
is that it actually has NOTHING to do with infinity.
As the proof occurs in ZF, the axiom of infinity IS NOT used.
This is because the proof applies EQUALLY well TO FINITE sets.
No square array contains its own anti-diagonal,
as a row or as a column. That REALLY IS common sense.
It remains common-sensical even if you allow the side-length of
the square to be omega (or any other countable ordinal)
as opposed to a finite ordinal -- NOT that there is any common
sense relevant to understanding what a "countable ordinal"
(in general) is. But there is some relevant to understanding
what the simplest/smallest one (omega) is, since it is just the
collection of all those finite things, all of which WERE available
to common sense.
george
science forum Guru Wannabe

Joined: 01 May 2005
Posts: 103

Posted: Thu Jul 13, 2006 5:51 pm    Post subject: Re: Question about Cantor's proof

david petry wrote:
 Quote: As such, mathematics is scientific in nature -- it has testable consequences. But Cantorian set theory, which was created to provide a formal basis for proving Cantor's theorem, leads to "mathematics" with no testable consequences.

You have absolutely no idea what "testable consequences"
means. You have tried to allege that |a^2 -2b^2|>=1
is a hypothesis that has "testable consequences",
while "the square root of 2 is irrational" is not one.

How can either of these have more or fewer "testable
consequences" than "the powerset of a set is bigger
than the set, EVEN when the set is infinite?"

How can you expect to be taken seriously?
george
science forum Guru Wannabe

Joined: 01 May 2005
Posts: 103

Posted: Thu Jul 13, 2006 6:01 pm    Post subject: Re: Question about Cantor's proof

 Quote: Cantor proving that the cardinality of reals is greater than the cardinality of integers is fine, it's a nice mathematical result which has no application in the physical world.

What UTTER bullshit.

For the nth time: Cantor's Theorem DOES NOT MENTION
infinity. The theorem is that the powerset is bigger than the
set, and the proof is by diagonalization.

The powerset of a set is the set of all its subsets.
If the set is countable to begin with then you can
make a LIST of (at least some of) the subsets.
Then you can ask how long this list is.

The theorem says that NO list whose length is the
same as the number of elements in the set (i.e. no
SQUARE array of bits, where the side-length is the
size of the set) contains ALL the subsets.
We can prove this simply by taking the anti-diagonal
of the square. It is guaranteed to disagree in (at least)
one place, with EVERY row (and EVERY column) of the
square, so it is NOT one of the subsets listed.

That is all.

This DOES SO TOO have physical consequences;
you can take 30 coins (Heads or Tails) and arrange 25 of them in a 5x5
square and use this theorem to prove that there has to be
an arrangement of them that does NOT occur as a row or
column of your square. You can then take the remaining
5 coins and arrange THEM in this HT pattern that did NOT
appear in the square. Knowing that you can always do this
is nice to know.

Thinking that somehow infinity makes it different is, well,
crazy.

 Quote: To feed people or send spacecraft to Mars you need the cybernetic interpretation of infinity:

This is just idiotic. However these finally wind up getting done,
I am quite confident it will NOT involve ANYbody asking YOU about
cybernetic infinity.

 Quote: The potential infinity has real applications, such as programming a spacecraft to report data "forever", knowing that some physical failure will evertually stop it. The actualized infinity is a mathematical concept only.

A mathematical concept that was absolutely necessary to
the discorver of the math that was used to compute the trajectory
of the data-reporter. And just for the record, it is NOT known that
"some physical failure will eventually stop it". Strange encounters
may be possible out there. Even if they are not, YOU don't know it.

 Quote: Cantor's nightmarish infinity of infinities must have been infinitey stressful for him, for he committed suicide.

The crank-posters here are slowly doing the same thing to themselves;
life is short, and they are wasting it, one post at a time. I can only
wonder whether we are not dragging ourselves down with them.
Jiri Lebl
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

Posted: Thu Jul 13, 2006 6:17 pm    Post subject: Re: Question about Cantor's proof

george wrote:
 Quote: Jiri Lebl wrote: Try going to a local bar and see how many people you convince with your proof. Hmph -- try going to a bar and seeing how many people remember what the concept of an infinite decimal is. Most people never understood infinity right THE FIRST time.

Thank you for having made my point (or something epsilon close to my
original point). I see no point in further discussing what you or I
mean by "common sense." We obviously have a very different definition
of those two words.

Jiri
David R Tribble
science forum Guru

Joined: 21 Jul 2005
Posts: 1005

Posted: Thu Jul 13, 2006 6:22 pm    Post subject: Re: Question about Cantor's proof

Aatu Koskensilta wrote:
 Quote: Cantor seems to have suffered of severe clinical depression, but there is no reason to suppose his illness had anything much to do with his mathematics or the opposition it met.

David R Tribble wrote:
 Quote: The irrational backlash from many of the mainstream mathematicians of the time and their efforts to slander him could not have helped his depression.

Aatu Koskensilta wrote:
 Quote: What efforts to slander Cantor are you talking about?

My mistake. The two signs of old age are loss of memory.

 Quote: From Wikipedia: It was once thought that Cantor's recurring bouts of depression were

triggered by the opposition his work met at the hands of Kronecker.
While Cantor's mathematical worries and his difficulties dealing with
certain people were greatly magnified by his depression, it is
doubtful whether they were its cause, which was probably bipolar
disorder.
george
science forum Guru Wannabe

Joined: 01 May 2005
Posts: 103

Posted: Sat Jul 15, 2006 4:14 am    Post subject: Re: Question about Cantor's proof

 Quote: george wrote: Hmph -- try going to a bar and seeing how many people remember what the concept of an infinite decimal is. Most people never understood infinity right THE FIRST time.

Jiri Lebl wrote:
 Quote: Thank you for having made my point (or something epsilon close to my original point). I see no point in further discussing what you or I mean by "common sense." We obviously have a very different definition of those two words.

You're being ridiculous.
What you are saying here is that infinity in general (or rather,
in the specific smallest case, the first case, the usual case)
is poorly understood by common sense. We could AGREE
on that. We would NOT have differing opinions about common
sense.

What you cut was my explanation that Cantor's Theorem
HAS NOTHING WHATSOEVER TO DO with infinity, so
claiming that it exceeds common sense, because it is about
diagonals
of squares of bit-strings. If people think that 0s and 1s surpass
their common sense then you can try heads & tails.
Jiri Lebl
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

Posted: Sat Jul 15, 2006 3:21 pm    Post subject: Re: Question about Cantor's proof

george wrote:
 Quote: What you cut was my explanation that Cantor's Theorem HAS NOTHING WHATSOEVER TO DO with infinity, so claiming that it exceeds common sense, because it is about infinity, is stupid. It is NOT about infinity. It IS about diagonals of squares of bit-strings. If people think that 0s and 1s surpass their common sense then you can try heads & tails.

That's because what you wrote was total nonsense. You CANNOT
generalize to infinity. You cannot just say we have "squares"
(whatever that means when the sides are infinite). With the "intuitive
reasoning" you used you may have arrived at the correct answer, but the
similar intuitive reasoning was used by Tono Orlow to show some total
nonsense.

Also I was talking about Cantor's theorem on uncountability of the
reals, not his theorem on |P(S)| > |S|. But even if we talk about that
theorem, yes it is obvious and intuitive for finite sets, but it is FAR
from common sense when we put infinite sets into the picture. You
simply CANNOT argue that you have a "square." I didn't say try the
proof for finite sets in a bar. Try it for infinite sets. You say the
proof is the same, it is NOT! It is the same only if you reason as you
have about "size" of the side of the "square." You'd have to first
explain cardinality concept which is not exactly obvious. Then try to
explain why evens and odds have the same cardinality and why if you
take every second line in your "square" you still get a square.

There is a reason why uncountability of the reals is not taught in say
primary school or high school.

Jiri
John Schutkeker
science forum Guru Wannabe

Joined: 30 May 2005
Posts: 172

Posted: Mon Jul 17, 2006 9:55 pm    Post subject: Re: Question about Cantor's proof

"Amanda" <sca18@hotmail.com> wrote in news:1152648112.752183.237420

 Quote: Why do so many people keep denying Cantor's theorem about the uncountability of the reals?

Why do people deny relativity, evolution or the moon landing?
Jonathan Hoyle
science forum Guru Wannabe

Joined: 04 Sep 2005
Posts: 260

Posted: Tue Jul 18, 2006 2:33 am    Post subject: Re: Question about Cantor's proof

 Quote: Why do so many people keep denying Cantor's theorem about the uncountability of the reals? Why do people deny relativity, evolution or the moon landing?

Well stated.

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