Towards a Paraconsistent Quantum Set Theory

TLDR: In this paper, a connection between quantum set theory and topos quantum theory was established by Ozawa, Takeuti and Titani, who studied algebraic valued set-theoretic structures whose truth values correspond to clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space.

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.