论文标题
改革Takeuti的量子设定理论以满足De Morgan的定律
Reforming Takeuti's Quantum Set Theory to Satisfy De Morgan's Laws
论文作者
论文摘要
1981年，Takeuti通过构建类似于布尔值模型的布尔逻辑的模型，引入了基于量子逻辑的集合理论。他为设定理论的每个句子定义了量子逻辑真实价值。他表明，平等公理不存在，而ZFC集合理论的公理如果用换向因子的概念进行了适当的修改，则保持理论。在这里，我们考虑了Takeuti的量子集理论中的问题，即De Morgan的定律不适合有限的量词。我们在Takeuti的量子集理论中构建了De Morgan定律的反示例。我们重新定义了成员关系和有限的生存量化的真实价值，以确保De Morgan的法律持有。然后，我们表明，ZFC集合理论的每个定理的真实价值都被其中的常数换入量子传递原理。
In 1981, Takeuti introduced set theory based on quantum logic by constructing a model analogous to Boolean-valued models for Boolean logic. He defined the quantum logical truth value for every sentence of set theory. He showed that equality axioms do not hold, while axioms of ZFC set theory hold if appropriately modified with the notion of commutators. Here, we consider the problem in Takeuti's quantum set theory that De Morgan's laws do not hold for bounded quantifiers. We construct a counter-example to De Morgan's laws for bounded quantifiers in Takeuti's quantum set theory. We redefine the truth value for the membership relation and bounded existential quantification to ensure that De Morgan's laws hold. Then, we show that the truth value of every theorem of ZFC set theory is lower bounded by the commutator of constants therein as quantum transfer principle.
