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### Abstract

This paper continues the study of the use of different models of ZF set theory as carriers for the mathematics of quantum mechanics. The basic tool used here is the construction of Cohen extensions of ZFC models by use of Boolean valued ZFC models [C=axiom of choice]. Let M be a standard transitive ZFC model. Inside M, *B* (H_{ M }) is the algebra of all bounded linear operators over some Hilbert spaceH_{ M }. It is shown that with each state ρ in *B* (H_{ M }) and projection operator ο in *B* (H_{ M }) one can associate a unique Boolean valued ZFC model M^{B} _{ρο}. B_{ρο} is the algebra of all Borel subsets of {0,1}^{ω}, the set of all infinite 0–1 sequences, modulo sets of *P* _{ρο} =⊗*p* _{ρο} measure zero with *p* _{ρο}({1}) =Trρο in M. Let Ψ_{ M } and Φ_{ M } be respective maps from the sets of state preparation and question measuring procedures into *B* (H_{ M }). Let M=M_{0}, the minimal standard transitive ZFC model. It is then shown that with each state preparation procedure *s* ‐ Dom(Ψ_{ M } _{ 0 }) and each question measuring procedure *q* ‐ Dom(Φ_{ M } _{ 0 }) and with each infinite repetition (*t* *s* *q*) of doing *s* and *q* at times *t* (0), *t* (1),..., if the definition of randomness is sufficiently strong, one can associate the Cohen extension M_{0}[ψ_{ t s q }] of M_{0} by ψ_{ t s q }. ψ_{ t s q } is the random outcome sequence associated with (*t* *s* *q*). A third condition, in addition to the two given in the previous paper, is then given which must be satisfied if a ZFC model M is to serve as a carrier for the mathematics of quantum mechanics. In essence it says that for each pair (*t* *s* *q*) and (*w* *u* *k*) of distinct infinite repetitions of doing *s* and *q* and of doing *u* and *k* with *s*, *u* ‐ Dom(Ψ_{ M }) and *q*, *k* ‐ Dom(Φ_{ M }), the two outcome sequences ψ_{ t s q } and ψ_{ w u k } are mutually statistically independent. It is then shown that for a strong definition of independence, corresponding to the definition of randomness used previously, no Cohen extension M_{0}[ψ_{ t s q }] of M_{0} can serve as the carrier for the mathematics of quantum mechanics.

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