Banach Tarski Paradox
Mike Jacobs
pieces are put together to form an object of larger volume; a marble can
be broken into pieces to form a sphere the size the sun. Is this a
correct statement of the paradox ?If so is there an intuitive expanation?
Tangent60
Well, yes, sort of. The Banach-Tarski theorem doesn't work in real life,
though, because there are only finitely many indivisible elements of matter
(definitely not uncountably many), namely atoms, leptons, or whatever. In the
mysterious, abstract world of analysis, there are uncountably many points,
which makes the Banach-Tarski theorem possible. Now, here's an inuitive (in my
opinion!) explanation:
Say a sphere is cut up into a finite number of pieces to form another, larger
sphere upon reunion of the pieces. Now, the pieces DON'T ALL HAVE TO BE
PATH-CONNECTED CHUNKS. They can be incomprehensibly weird and "discontinuous"
sets of points that, upon reunion, use the property of infinity to make the
sphere larger. (I hope that didn't confuse anyone.) In fact, I am currently in
the belief that the reason everyone believes this is a paradox because of their
misinterpretation of infinity (which is annoying to all of us and is found in
every single human. Sigh...).
Mike Oliver
Tangent60 wrote:
> Now, the pieces DON'T ALL HAVE TO BE
> PATH-CONNECTED CHUNKS.
I have a memory of seeing written somewhere that the pieces can be
chosen to be both connected and locally path-connected. Does anyone
know whether this is true?
> In fact, I am currently in the belief that the reason everyone
> believes this is a paradox because of their misinterpretation of
> infinity [....]
Oh, there's no question that it's a paradox. It's a paradox you
can get used to, so that it no longer seems paradoxical ... but
that's true of all paradoxes, intuition being the highly trainable
thing that it is.
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Mike Jacobs
elements of matter and we know this is not the case for real matter then
it is not a paradox. It is just a waste of time.
I read in the book by Gibbons "Shrodingers Kittens etc etc" where he
spends a few pages spouting out that there are profound physical
consequencesof the BTT in quantum theory .That is too incredible to be
true. Thank you Mr Tangent for your response.
Mike McCarty
about matter, and in fact has nothing to do with matter.
Doesn't mean it's a waste of time, though.
Mike
In article <71uij6$1gma$1...@newssvr03-int.news.prodigy.com>,
Mike Jacobs <CBR...@prodigy.com> wrote:
)If the theorem assumes that there are an infinite number of individual
)elements of matter and we know this is not the case for real matter then
)it is not a paradox. It is just a waste of time.
) I read in the book by Gibbons "Shrodingers Kittens etc etc" where he
)spends a few pages spouting out that there are profound physical
)consequencesof the BTT in quantum theory .That is too incredible to be
)true. Thank you Mr Tangent for your response.
)
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Mike Jacobs
is to short to worry about hypothetical mathematical constructs.
Tangent60
For a minute, I'd say you were a mathematician.
>Life is to short to worry about hypothetical mathematical constructs.
But...
Mike Jacobs
Try it youll like it.
Tangent60
> Try it youll like it.
The same thing happened to me, just in the opposite direction. I was very
interested in physics at one point, but then I _gradually_ slipped into
mathematics. I liked it much more. But then again, I didn't _really_ do
physics, like the majors do. Perhaps when I get all the right math knowledge
(on the side of infinitesimals and tensor calculus, which I may learn only for
the purpose of this or that theory..) I could dabble for a while in physics,
and see how it is. :)
Nils Chr. Framstad
>
> I was a math person until I saw the light. Physics is so much more fun.
> Try it youll like it.
I wanted to study physics until I met this four-letter word named "lab"
:)
--
mailto: n...@math.uio.no
_ _ _ _
If pi were equal to 3, this sentence w<_>uld l<_><_>k s<_>mething like
this. (cf. 1Kings7:23 - http://home.sol.no/~ggunners/bibel/1kg/7.htm)
Mike Jacobs
learned some pre 20th century math I went over to to Physics. The
rest of the stuff gives me a headache. Post 20th century physics on the
other hand is intensely interesting. Read Feynman if you want more
philosopy on this subject. But in any case have fun.
Tangent60
Oh, yeah. I remember that now...one less reason for me to study physics - the
boring part. :(
Michael Wodzak
>
> After I found out how to generate the real numbers from set theory and
> learned some pre 20th century math I went over to to Physics. The
> rest of the stuff gives me a headache. Post 20th century physics on the
> other hand is intensely interesting.
Ah ah! so the true turn of the Millenium came in or before 1998. I'm
glad we can lay that controversy to rest! ;o)
Michal
Ron Rood
Nils Chr. Framstad wrote:
> I wanted to study physics until I met this four-letter word named "lab"
What is this four-letter word named "lab"?
Ron
Mike McCarty
Ron Rood <"P.T.M. Rood"@th.vu.nl> wrote:
)
)
)Nils Chr. Framstad wrote:
)
)> I wanted to study physics until I met this four-letter word named "lab"
)
)What is this four-letter word named "lab"?
It is "lab". Actually, he left off a layer of quotes. He should have
written
I wanted to study physics until I met this four-letter word
named "'lab'".
Mike
Michael Wodzak
>
> It is "lab". Actually, he left off a layer of quotes. He should have
> written
>
> I wanted to study physics until I met this four-letter word
> named "'lab'".
Isn't it a six letter word??? ;o)
Ronald Bruck
Mike McCarty <jmcc...@sun1307.spd.dsccc.com> wrote:
:In article <3649482A...@th.vu.nl>,:)> I wanted to study physics until I met this four-letter word named "lab"
:)
:)What is this four-letter word named "lab"?
:
:It is "lab". Actually, he left off a layer of quotes. He should have
:written
:
: I wanted to study physics until I met this four-letter word
: named "'lab'".
I wasn't sure whether this question wasn't a subtle joke, a la Alice,
talking to the White Rabbit? not sure which, distinguishing between a
thing, a thing's NAME, and what the thing is CALLED.
So the four-letter word could be NAMED "lab" yet be some other word ;-)
More likely, I think it was in the genre of the old saw, "There are three
kinds of mathematicians; those that can count, and those that can't." A
joke.
--Ron Bruck
Bill Taylor
|>
Thank you Lewis.
I think he meant that it is called "lab".
But we still don't know what its name is called!
Anyway, he's right, physics labs are a ####!
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
We should all try to avoid the "c" word and the "f" word. (chaos & fractals)
-------------------------------------------------------------------------------
bo...@rsa.com
Nope. Your response seems to indicate you missed the joke.
"four letter word" = "dirty word"
The jokes says that "lab" is a dirty word.
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Mike McCarty
Ronald Bruck <br...@math.usc.edu> wrote:
)In
)Mike McCarty <jmcc...@sun1307.spd.dsccc.com> wrote:
):In article <3649482A...@th.vu.nl>,):)
):)Nils Chr. Framstad wrote:
):)
):)> I wanted to study physics until I met this four-letter word named "lab"
):)
):)What is this four-letter word named "lab"?
):It is "lab". Actually, he left off a layer of quotes. He should have
):written
):
): I wanted to study physics until I met this four-letter word
)
)I wasn't sure whether this question wasn't a subtle joke, a la Alice,
)talking to the White Rabbit? not sure which, distinguishing between a
)thing, a thing's NAME, and what the thing is CALLED.
)
)So the four-letter word could be NAMED "lab" yet be some other word ;-)
Um, that's possible, in which case he didn't need the second layer of
inverted commas.
)More likely, I think it was in the genre of the old saw, "There are three
)kinds of mathematicians; those that can count, and those that can't." A
)joke.
That's the way I took it, in which case he missed a layer of quotation.
IOW, he needed to put the name in quotes. The name is then "lab", and to
put the name itself in for reference, another layer is needed "'lab'".
Mike McCarty
)In article <72dtrb$r2e$1...@math.usc.edu>,
) br...@math.usc.edu (Ronald Bruck) wrote:
)> In
)> Mike McCarty <jmcc...@sun1307.spd.dsccc.com> wrote:
)> :In article <3649482A...@th.vu.nl>,
)> :Ron Rood <"P.T.M. Rood"@th.vu.nl> wrote:
)> :)
)> :)
)> :)Nils Chr. Framstad wrote:
)> :)
)> :)> I wanted to study physics until I met this four-letter word named "lab"
)> :)
)> :)What is this four-letter word named "lab"?
)> :
)> :It is "lab". Actually, he left off a layer of quotes. He should have
)> :written
)> :
)> : I wanted to study physics until I met this four-letter word
)> : named "'lab'".
)>
)> talking to the White Rabbit? not sure which, distinguishing between a
)> thing, a thing's NAME, and what the thing is CALLED.
)>
)> So the four-letter word could be NAMED "lab" yet be some other word ;-)
)>
)> More likely, I think it was in the genre of the old saw, "There are three
)> kinds of mathematicians; those that can count, and those that can't." A
)> joke.
)
)
)Nope. Your response seems to indicate you missed the joke.
)
)"four letter word" = "dirty word"
)
)The jokes says that "lab" is a dirty word.
That is the way I took it, and that is why he needed another layer of quotes.