From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Bayes' theorem and QM
Organization: ORA Corporation
Date: Wed, 6 Jan 1993 13:25:37 GMT

jgk@osc.COM (Joe Keane) writes:

>>It would be nice if the volume of a region of space were always
>>well-defined, but it is not. A demonstration due to Banach and Tarski
>>showed that it is possible (mathematically, rather than physically
>>possible) to decompose a sphere into a finite number of pieces and
>>then recombine them by rotations and translations to get two complete
>>spheres.

>Banach-Tarski is a good demonstration of the tricks you can play if
>you have axioms which claim the existence of objects with infinite
>information, or however you want to interpret the non-constructible
>things that you can get only with the Axiom of Choice. But does
>anyone think this has any relation to reality?

I think it would be useful for you to look at a proof of the
Banach-Tarski paradox before making a judgement about its relevance to
reality. Even without the axiom of choice, it is possible to do the
following:

     1. Start with a countable dense subset of the sphere.
     2. Divide this "holey" sphere into a finite number of pieces.
     3. Rearrange the pieces by rotations and translations so as to
        form 2 complete "holey" spheres equivalent to the original.

So, the trick involved in the Banach-Tarski paradox, dissecting an
object and reassembling it into a larger object, does not require
the axiom of choice. The axiom of choice is only necessary to extend
this result to an uncountable number of points: the entire sphere.

>If you think so, please show me some pieces that you could make out of
>something real like wood, such that that they almost fit together into
>either one sphere or two. Also please show me a computer chip that
>can store an infinite amount of information.  In fact i'll be
>generous, you only have to store an arbitrarily large amount of
>information.

Please read what I wrote:

"A demonstration due to Banach and Tarski showed that it is possible
(mathematically, rather than physically possible) to..."
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The point is not that we can follow the Banach Tarski construction and
multiply wood spheres, the point is that measure theory in which some
sets are nonmeasurable is consistent. Therefore, there is nothing
inherently contradictory about a probability theory that fails to
give probabilities to some meaningful events.

>Don't get me wrong, i'm quite interested in math logic.  I like
>considering strange axioms that let you enumerate the reals or decide
>whether Turing machines will halt.  (No, these are not contradictory,
>as long as you understand that such an axiom applies only to things
>you can make without using the axiom.)  But i'm worried if people
>think we can do new things in the real world by inventing new axioms
>about infinite sets.

You seem to think that mathematics is only about what "we" can do or
make. What do you mean by "we"? Of course, a mathematical proof of the
existence of an object does not imply *human* ability to construct
such an object. However, there is no a priori reason to think that
nature is limited to what humans are capable of doing.

Daryl McCullough
ORA Corp.
Ithaca, NY
Newsgroups: sci.math