[FOM] banach-tarski paradox

Monroe Eskew meskew at math.uci.edu
Fri Sep 25 00:22:50 EDT 2009

On Thu, Sep 24, 2009 at 10:55 AM,  <joeshipman at aol.com> wrote:
> I wonder whether people who understand the proof of the Banach-Tarski
> theorem are less likely to regard it as evidence against the Axiom of
> Choice than people who are merely informed that it is a consequence of
> the Axiom of Choice. My reaction to the proof is that is disproves the
> conjunction of the two physical principles "space is isotropic" and
> "matter is infinitely divisible" and hence represents a useful
> application of mathematics to physics, but does not make the Axiom of
> Choice less plausible.

I don't think you have to go that far.  Matter could be infinitely
divisible in the sense of not having a smallest part, a kind of
"potential infinity," while at the same time not allowing
Banach-Tarski cuts of a ball, or any other cuts requiring (1) an
"infinite knife" (if you will) and (2) the cut matter to be rigid in
certain ways.

I think we just have to realize that there the questions of (a) what
sets exist, and (b) what kinds of clay models can we actually make,
are very different.

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