Sheaf Logic, Quantum Set Theory and the Interpretation of Quantum Mechanics

TL;DR

This paper develops a sheaf logic-based framework for quantum set theory, providing new insights into quantum mechanics interpretation and the emergence of classicality through a constructive approach linked to the Deutsch-Everett multiverse.

Contribution

It introduces a hierarchy of Quantum Variable Sets using sheaf logic, simplifying prior models and connecting quantum set theory with the Gelfand representation and multiversal interpretations.

Findings

Constructs a hierarchy of Quantum Variable Sets via sheaf logic.

Provides alternative, more constructive proofs of the correspondence between operators and real numbers.

Links the model to the emergence of classicality in quantum mechanics.

Abstract

Based on the Sheaf Logic approach to set theoretic forcing, a hierarchy of Quantum Variable Sets is constructed which generalizes and simplifies the analogous construction developed by Takeuti on boolean valued models of set theory. Over this model two alternative proofs of Takeuti's correspondence, between self adjoint operators and the real numbers of the model, are given. This approach results to be more constructive showing a direct relation with the Gelfand representation theorem, revealing also the importance of these results with respect to the interpretation of Quantum Mechanics in close connection with the Deutsch-Everett multiversal interpretation. Finally, it is shown how in this context the notion of genericity and the corresponding generic model theorem can help to explain the emergence of classicality also in connection with the Deutsch- Everett perspective.