There is No Axiomatic System for the Quantum Theory

We show that there is a contradiction within quantum mechanics. We derive a proposition concerning a quantum expectation value under the assumption of the existence of the directions in a spin-1/2 system. The quantum predictions within the formalism of von Neumann's projective measurement cannot coexist with the proposition concerning the existence of the directions. Therefore, we have to give up either the existence of the directions or the formalism of von Neumann's projective measurement. Hence there is a contradiction within the Hilbert space formalism of the quantum theory. This implies that there is no axiomatic system for the quantum theory. We need new physical theories in order to explain mathematically the handing of raw experimental data.