对Takeuti量子集合论的改进以满足De Morgan定律
Reforming Takeuti's Quantum Set Theory to Satisfy De Morgan's Laws
摘要：
1981年，Takeuti通过构建类似于布尔逻辑的布尔值模型，引入了基于量子逻辑的集合论。他为集合论的每一句话定义了量子逻辑真值。他表明，平等公理并不成立，而公理的ZFC集理论举行，如果适当修改的概念，消除。在这里，我们考虑Takeuti的量子集合论中的问题，即德摩根定律不适用于有界量词。我们构造了一个反例，以德摩根的法律有界量词在Takeuti的量子集合论。我们重新定义了隶属关系和有界存在量化的真值，以确保德摩根定律成立。然后，我们证明了ZFC集合论的每个定理的真值都是由其中的常数交换子作为量子转移原理的下界。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
