Models of Zermelo Frankel set theory as carriers for the mathematics of physics. I

Paul A.  Benioff

doi:10.1063/1.522953

Abstract

This paper is a first attempt to explore the relationship between physics and mathematics ’’in the large.’’ In particular, the use of different Zermelo Frankel model universes of sets (ZFC models) as carriers for the mathematics of quantum mechanics is discussed. It is proved that given a standard transitive ZFC model M, if, inside M, B (H_M) is the algebra of all bounded linear operators over a Hilbert space

H

_M, there exists, outside M, a Hilbert space H and an algebra B (H), along with isometric monomorphisms U_M and V_M from H_M into H and from B (H_M) into B (H). U_M and V_M are used to relate quantum mechanics based on M to quantum mechanics based on the usual ZFC model. It is then shown that, contrary to what one would expect, all ZFC models may not be equivalent as carriers for the mathematics of physics. In particular, it is proved that if one requires that an outcome sequence, associated with an infinite repetition of measuring a question observable on a system prepared in some state, be random, and if a strong definition of randomness is used, then the minimal standard ZFC model cannot be a carrier for the mathematics of quantum mechanics.

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Models of Zermelo Frankel set theory as carriers for the mathematics of physics. I

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