Orthomodular-Valued Models for Quantum Set Theory

TL;DR

This paper unifies quantum and Boolean set theories using orthomodular lattices, generalizes the transfer principle, and explores implications for quantum logic models, enhancing flexibility in quantum set theory frameworks.

Contribution

It introduces generalized implications extending quantum logic, proves the transfer principle for these, and broadens the scope of quantum set theory models beyond the Sasaki arrow.

Findings

Generalized transfer principle holds for all studied implications.

All polynomially definable implications satisfying the transfer principle are identified.

The approach allows abandoning the Sasaki arrow, increasing model flexibility.

Abstract

In 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo-Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti's model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even non-polynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti's model and leads to a much more flexible approach to quantum set theory.