[FOM] banach-tarski paradox

joeshipman@aol.com joeshipman at aol.com
Thu Sep 24 13:55:59 EDT 2009

That article by Maddy ("How Applied Mathematics Became Pure", The 
Review of Symbolic Logic v.1 #1 (June 2008) pp. 16-41) is quite 
interesting and is about much more than ZFC (it can be found online at 
). Caution: the article contains a repeated, extremely nonstandard use 
of the term "the continuum hypothesis" to refer to a statement about 
the physical applicability of continuum mechanics.

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.

-- JS

-----Original Message-----
From: Monroe Eskew <meskew at math.uci.edu>

In a recent paper, Penelope Maddy discusses what the Banach-Tarksi
paradox has to say about the empirical truth of ZFC.  She says that
the existence claim of Banach-Tarski "seems obviously absurd from a
physical point of view," and notes that some have taken it as
empirical evidence against Choice.  As an alternative response, she
suggests instead concluding that the Lebesgue measurable subsets of
R^3, or perhaps a subset of L(R), is a better model of physical space.

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