Orthomodular-Valued Models for quantum Set Theory

Orthomodular logic represented by a complete orthomodular lattice has been studied as a pertinent generalization of the two-valued logic, Boolean-valued logic, and quantum logic. In this paper, we introduce orthomodular logic valued models for set theory generalizing quantum logic valued models introduced by Takeuti as well as Boolean-valued models introduced by Scott and Solovay, and prove a general transfer principle that states that every theorem of ZFC set theory without free variable is, if modified by restricting every unbounded quantifier appropriately with the notion of commutators, valid in any orthomodular logic valued models for set theory. This extends the well-known transfer principle for Boolean-valued models. In order to overcome an unsolved problem on the implication in quantum logic, we introduce the notion of generalized implications in orthomodular logic by simple requirements satisfied by the well-known six polynomial implication candidates, and show that for every choice from generalized implications the above transfer principle holds. In view of the close connection between interpretations of quantum mechanics and quantum set theory, this opens an interesting problem as to how the choice of implication affects the interpretation of quantum mechanics.