Towards a Paraconsistent Quantum Set Theory

TL;DR

This paper explores the connection between quantum set theory and topos quantum theory by constructing algebraic set-theoretic structures with truth values linked to spectral presheaves, aiming to unify different quantum logical frameworks.

Contribution

It introduces algebraic valued set-theoretic structures that relate quantum set theory to topos quantum theory, and attempts to recreate Takeuti's isomorphism within these new models.

Findings

Established a link between quantum set theory and topos quantum theory.

Constructed algebraic structures with truth values based on spectral presheaves.

Recreated Takeuti's isomorphism in the new set-theoretic models.

Abstract

In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani, and topos quantum theory, as developed by Isham, Butterfield and Doring, amongst others. Towards this end, we will study algebraic valued set-theoretic structures whose truth values correspond to the clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space. In particular, we will attempt to recreate, in these new structures, Takeuti's original isomorphism between the set of all Dedekind real numbers in a suitably constructed model of set theory and the set of all self adjoint operators on a chosen Hilbert space.