  It is usually stated that quantum mechanics presents problems with the
identity of particles, the most radical position -supported by E. Schrodinger-
asserting that elementary particles are not individuals. But the subject goes
deeper, and it is even possible to obtain states with an undefined particle
number. In this work we present a set theoretical framework for the description
of undefined particle number states in quantum mechanics which provides a
precise logical meaning for this notion. This construction goes in the line of
solving a problem posed by Y. Manin, namely, to incorporate quantum mechanical
notions at the foundations of mathematics. We also show that our system is
capable of representing quantum superpositions.
