Quantum Set Theory and Foundations of Quantum Mechanics

Since the introduction by Birkhoff and von Neumann, quantum logic has been considered a fundamental tool to investigate foundations of quantum mechanics. However, the later developments have shown a certain weakness. For foundations of mathematics, it is well-known that propositional logic is too weak but set theory has a complete power. Thus, we can accept the weakness of quantum logic as a quantum version of propositional logic but we can expect a full power of quantum set theory for foundations of quantum mechanics. Quantum set theory was instituted by Gaishi Takeuti in 1981 as a modification of Boolean-valued models of set theory adapted to quantum logic but has not been called for attention for more than 20 years, mainly because of technical difficulty. Here, I will report my recent investigations on quantum set theory and show how rich is the role of quantum set theory in foundations of quantum mechanics. [1] M. Ozawa, Transfer principle in quantum set theory, J. Symbolic Logic 72, 625-648 (2007). [2] M. Ozawa, Orthomodular-valued models for quantum set theory, arXiv:0908.0367