  Intuitively, the more powerful a theory is, the greater the variety and
quantity of ideas can be expressed through its formal language. Therefore, when
comparing two theories concerning the same subject, it seems only reasonable to
compare the expressive powers of their formal languages. On condition that the
quantum mechanical description is universal and so can be applied to
macroscopic systems, quantum theory is required to be more powerful than
classical mechanics. This implies that the formal language of Hilbert space
theory must be more expressive than that of Zermelo-Fraenkel set theory (the
language of classical formalism). However, as shown in the paper, such a
requirement cannot be met. As a result, classical and quantum formalisms cannot
be in a hierarchical relation, that is, include one another. This fact puts in
doubt the quantum-classical correspondence and undermines the reductionist
approach to the physical world.
