  The theory of quantum mechanics is examined using non-standard real numbers,
called quantum real numbers (qr-numbers), that are constructed from standard
Hilbert space entities. Our goal is to resolve some of the paradoxical features
of the standard theory by providing the physical attributes of quantum systems
with numerical values that are Dedekind real numbers in the topos of sheaves on
the state space of the quantum system. The measured standard real number values
of a physical attribute are then obtained as constant qr-number approximations
to variable qr-numbers. Considered as attributes, the spatial locations of
massive quantum particles form non-classical spatial continua in which a single
particle can have a quantum trajectory which passes through two classically
separated slits and the two particles in the Bohm-Bell experiment stay close to
each other in quantum space so that Einstein locality is retained.
