FOM: Mathematics and Physics -- some details
jshipman@bloomberg.net
jshipman at bloomberg.net
Wed Nov 19 09:04:46 EST 1997
I misstated my own results in an earlier posting. Itamar Pitowsky and
Stanley Gudder reformulated Quantum Mechanics as a "local hidden variables"
theory. This did not violate Bell's theorem which says such theories are
incompatible with QM because they cleverly skirt the axioms of conditional
probability by using nonmeasurable sets in an essential way. (By the quantum
mechanical principle of conservation of weirdness this does not resolve all the
philosophical problems, but replaces a physical weirdness (spooky action at a
distance) with a mathematical, Banach-Tarski paradox kind of weirdness.) The
key technical point is that measurement of noncommuting observables in different
orders corresponds to iterating integrals in different orders--by Fubini's
theorem this can only give different answers for non-measurable functions. I
showed ("Cardinal conditions for strong Fubini theorems", T.A.M.S. 10/90) that
the existence of their nonmeasurable functions was independent of ZFC. They'd
used CH to get non-Fubini functions (Martin's axiom would also work); I showed
it was consistent with ZFC (and implied by RVM) that there were none.--J Shipman
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