量子集合论中的转移原理
Transfer principle in quantum set theory
摘要：
在1981年，Takeuti引入量子集合论作为布尔值模型的量子对应物的集合论通过构建一个模型的集合论的基础上量子逻辑表示的格子封闭子空间在希尔伯特空间，并表明适当的量子对应物的ZFC公理举行的模型。在这里，Takeuti的配方是扩展到构建一个模型的集合论的基础上表示的逻辑格的投影在任意冯诺依曼代数。建立了一个转移原理，使我们能够将ZFC定理转移到模型中的量子对应定理。模型中的实数集与逻辑生成的冯诺依曼代数中的自伴算子集一一对应。尽管Takeuti指出的困难，平等公理一般不持有量子集合论，它表明，平等公理持有任何实数的模型。证明了量子力学中的任何观测命题都可以用模型中实数的对应陈述来表示，其真假值与量子力学的标准表述一致，并且模型中两个实数之间的等式关系与量子力学中相应观测量（自伴算子）之间的完全相关概念等价.本文最后对量子逻辑中蕴涵联结词的选择与量子集合论的关系作了一些评论。
英文摘要：
In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.
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