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.

TeX-file, 32 pages