  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.
