Quantum Set Theory
Quantum Set Theory
复制标题
DOI：
10.1007/978-1-4613-3228-2_19
复制
发表时间：
1981
期刊：
影响因子：
--
通讯作者：
G. Takeuti
中科院分区：
文献类型：
--
作者：
G. Takeuti
本文研究了基于量子逻辑的集合论。量子逻辑是指希尔伯特空间的所有闭线性子空间的格。由于量子逻辑是一种内在逻辑，即量子世界的逻辑，（参见1）发展基于量子逻辑的数学，更具体地说，基于量子逻辑的集合论是一个重要的问题。这也是一个具有挑战性的问题，因为量子逻辑与经典逻辑或直觉主义逻辑有很大的不同，因此基于量子逻辑的数学非常困难。另一方面，以量子逻辑为基础的数学具有非常丰富的数学内容。量子逻辑中有许多完整的布尔代数，这一事实清楚地表明了这一点。对于每一个完备布尔代数B，基于B的数学已经被我们关于布尔值分析的工作所证明，具有丰富的数学意义。由于基于B的数学可以被认为是基于量子逻辑的数学的一个子理论，所以毫无疑问，基于量子逻辑的数学是非常丰富的。情况似乎如下。以量子逻辑为基础的数学太庞大了，难以看清。
In this paper, we study set theory based on quantum logic. By quantum logic, we mean the lattice of all closed linear subspaces of a Hilbert space. Since quantum logic is an intrinsic logic, i.e. the logic of the quantum world, (cf.1) it is an important problem to develop mathematics based on quantum logic, more specifically set theory based on quantum logic. It is also a challenging problem for logicians since quantum logic is drastically different from the classical logic or the intuitionistic logic and consequently mathematics based on quantum logic is extremely difficult. On the other hand, mathematics based on quantum logic has a very rich mathematical content. This is clearly shown by the fact that there are many complete Boolean algebras inside quantum logic. For each complete Boolean algebraB, mathematics based onBhas been shown by our work on Boolean valued analysis4, 5, 6to have rich mathematical meaning. Since mathematics based onBcan be considered as a sub-theory of mathematics based on quantum logic, there is no doubt about the fact that mathematics based on quantum logic is very rich. The situation seems to be the following. Mathematics based on quantum logic is too gigantic to see through clearly.