[FOM] banach-tarski paradox
Monroe Eskew
meskew at math.uci.edu
Thu Sep 24 03:16:00 EDT 2009
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.
Perhaps a better response would be that the inability to physically
cut an object in a certain way does say much about the existence of
subsets of space. The dynamical laws and the constitution of physical
objects may not allow a certain set to be displayed as the exact
region occupied by piece of matter, even if that set exists. For
example, it seems impossible to make a physical object corresponding
exactly to the set of rational points in the unit cube, or the Cantor
set, yet both these sets are in L(R). The proper response seems to be
to distinguish between set-theoretic ontology and the laws governing
matter.
Best,
Monroe
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