  Thought experiments about the physical nature of set theoretical
counterexamples to the axiom of choice motivate the investigation of peculiar
constructions, e.g. an infinite dimensional Hilbert space with a modular
quantum logic. Applying a concept due to BENIOFF, we identify the intrinsically
effective Hamiltonians with those observables of quantum theory which may
coexist with a failure of the axiom of choice. Here a self adjoint operator is
intrinsically effective, iff the Schroedinger equation of its generated
semigroup is soluble by means of eigenfunction series expansions.
