The formulation of quantum physics in terms of lattice theory, as introduced by Birkhoff and von Neumann, represents a system of quantum physics as a Hilbert space whose elements correspond to physical states while propositions correspond to closed subspaces of the Hilbert space. These subspaces form a complete orthomodular lattice which may be considered as a quantum logic. \textit{G. Takeuti} in [Quantum Set Theory, Current Issues in Quantum Logic], \textit{E. Beltrametti} and \textit{B. C. van Frassen}, (eds.), (Plenum, New York), 303--322 (1981; Zbl 0537.03044)] developed a quantum set theory based on quantum logic. The authors continue research in this direction. They formulate the quantum set theory by introducing a strong implication corresponding to the lattice order, and represent the basic concepts of quantum physics such as propositions, symmetries, and states.