

The Banach–Tarski paradox - acangiano
http://en.wikipedia.org/wiki/Banach–Tarski_paradox

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jonshea
Q: What’s an anagram of “Banach-Tarski”? A: Banach-Tarski Banach-Tarski

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abstractbill
The most interesting thing about the BT paradox, for me, is its interplay with
the Axiom Of Choice. Most of the mathematics I did as part of my PhD would
fall apart without AC, but I still remember being very unsettled by BT.

 _The existence of nonmeasurable sets, such as those in the Banach–Tarski
paradox, has been used as an argument against the axiom of choice.
Nevertheless, most mathematicians are willing to tolerate the existence of
nonmeasurable sets, given that the axiom of choice has many other
mathematically useful consequences._

~~~
d0mine
The funny things is that the same result for an orbit on a sphere doesn't need
the Axiom Of Choice at all. It just uses the concept of actual infinity.

 _"the axiom of choice has many other mathematically useful consequences."_ is
an understatement. It's like a life in Space. You _can_ live in Space (in a
specially design isolated constructions e.g., a space station); you even can
go outside (not for long and only if you are wearing a space-suit) but It is
much much easier to live on the Earth.

~~~
tome
I'd be interested in seeing a reference for the result on the sphere, if you
have one.

~~~
d0mine
It is the theorem from a dead-tree math-textbook where it is used as a step in
a Banach-Tarski Paradox proving.

The textbook is not in English. Here's a translation of the theorem:

An orbit O [1] can be decomposed into 4 sets: A, B, C, D. Using rotation these
sets can be combined into 2 orbits:

    
    
      A ∪ aB = O; C ∪ bD = O
    

Proof:

    
    
      A = H(a)x; B = H(a')x; C = H(b)x; D = H(b')x
    

The theorem statement follows from the fact that the free group H can be
decomposed into 4 parts:

    
    
      H(a),H(a'),H(b),H(b')

and doubled by rotations:

    
    
      H = aH(a') ∪ H(a); H = bH(b') ∪ H(b)

∎

__ [1]: The term `orbit` is used in the same sense as in [2]. `H` is a free
group similar to the one from the step 3 in [2], and `a`, `b` are rotations
defined similar to Step 2 in [2] i.e., they are generators of H:

    
    
      H = {e}∪H(a)∪H(a')∪H(b)∪H(b')
    

, where `e` is the unit of the group H:

    
    
      aa' = e; bb' = e
    

__ [2]: <http://en.wikipedia.org/wiki/Banach–Tarski_paradox>

~~~
tome
I can't follow either of the proofs yet, but one question does pose itself:
given a decomposition of the sphere, don't you get a decomposition of the
ball, just by projecting towards the origin?

Of course what happens at the origin itself is not clear. Perhaps there's no
way of getting around that difficulty. If you could deal with that, however, a
proof on the sphere would be equivalent to a proof on the ball.

~~~
d0mine
An orbit is not a sphere. To get the result for a sphere we need AC in some
form.

The result for the sphere are easily generalized on a ball (first without a
center, then with the center).

------
ced
_Infinite set paradoxing has become a morbid infection that is today spreading
in a way that threatens the very life of probability theory, and requires
immediate surgical removal._ \--- E.T. Jaynes

It applies to other theories as well. Giving real-life meaning to a
construction based on infinity and then shouting "paradox!" seems counter-
productive.

------
Eliezer
"The Banach-Tarski Gyroscope is an intricate mechanism believed to have been
constructed using the Axiom of Choice. On each complete rotation
counterclockwise, the Banach-Tarski Gyroscope doubles in volume while
maintaining its shape and density; on rotating clockwise, the volume is
halved. When first discovered, fortunately in the midst of interstellar space,
the Banach-Tarski Gyroscope was tragically mistaken for an ordinary desk
ornament. Subsequently it required a significant portion of the available
energy of the contemporary galactic civilization to reverse the rotation
before nearby star systems were endangered; fortunately, the Banach-Tarski
Gyroscope still obeys lightspeed limitations on rotation rates, and cannot
grow rapidly once expanding past planetary size. After the subsequent
investigation, the Banach-Tarski Gyroscope was spun clockwise and left
spinning."

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dcminter
Isn't this the one that was mocked by Feynman?

~~~
acangiano
Yes, it is. Feynman was not a big fan of things that couldn't be proved
experimentally.

~~~
RiderOfGiraffes
Feynman's strength (well, one of them) was that he insisted on a simple
example to follow during an experiment. This is an incredibly powerful
technique for understanding complex calculations, and no doubt many hackers
here use it when doing a code walk-through.

The problem was that the mathematicians fell into the trap Feynman laid and
gave overly simple descriptions. Feynman seduced them into trying to give
physical descriptions when it was, in truth, important to think about the
details.

Banach-Tarski is a specific example of that, as many of you will know. I think
Feynman knew full well what he was doing - just one of the games he played.

The technique has its limitations too. Restricting yourself to consider only
the physical loses the power of abstraction. Many of you know that too,
although perhaps you don't think of it that way. When you extract a method
you're using abstraction - the method may not represent something physical.
Feynman's technique can limit you there.

The story comes from "Surely You're Joking, Mr Feynam", in the chapter called
"A Different Box of Tools."

[http://www.multitran.ru/c/m.exe?a=DisplayParaSent&fname=...](http://www.multitran.ru/c/m.exe?a=DisplayParaSent&fname=Richard%20Feynman%5CChapter12)

Feynman makes a big deal about having unusual tools in the box - they let you
solve problems that others find intractable. That's one real lesson. Acquire a
wide variety tools, and know how to use them. Functional, Object Oriented,
Imperative, Logic, Database, etc. They all have their place, and knowing them,
really knowing them, gives you enormous flexibility.

Never stop learning new tricks.

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gjm11
Stan Wagon's book about the B-T paradox is absolutely superb, but for
mathematicians only. I've heard good things about Wapner's "The pea and the
sun", intended for a lay audience -- e.g., the reviews at
<http://www.maa.org/reviews/PeaSun.html> and
<http://www.ams.org/notices/200609/rev-komjath.pdf> \-- but I haven't read it
myself.

(Speaking of Stan Wagon and beautiful mathematical exposition, his drably-
titled paper "Fourteen proofs of a result about tiling a rectangle" is lovely
and quite accessible. You can find it on the web with a bit of googling; I
don't know whether any copy you can find in this way is legal.)

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rms
Be careful, friends... digging too deep into set theory is one of the surest
ways to go insane. You quickly reach the edge of what you consciousness can
handle.

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randallsquared
Sort of a reductio ad absurdum proof against infinities? :)

~~~
RiderOfGiraffes
Well, against the Axiom of Choice, at least.

Attempts were once made to show that Euclid's fifth postulate could be deduced
from the others by assuming it was false, and seeing where that led. A
perfectly valid technique, as you know. Several assumed that it succeeded,
because the conclusions were so obviously ludicrous, and from that it was
assumed that Euclid's geometry was the only one.

Of course, now we know that the conclusions weren't contradictions, and are
"simply" non-Euclidean geometries.

"Counter-Intuitive" is not the same as "Contradiction."

Just because the Axiom of Choice leads to such counter-intuitive results it
doesn't mean they're not true. Just because infinities lead to bigger
infinities (via Cantor), flasks of finite volume but infinite surface area,

<http://community.tes.co.uk/forums/t/277351.aspx>

and hotels that never run out of room,

[http://mathforum.org/kb/message.jspa?messageID=5584228&t...](http://mathforum.org/kb/message.jspa?messageID=5584228&tstart=0)

it doesn't mean that they aren't useful, or don't, in some sense, exist.

