On the computational power of physical systems, undecidability, the consistency of phenomena and the practical uses of paradoxa

Just as in mathematics, recursion theory and formal logic paradoxa can be used to derive incompleteness theorems, physical paradoxa can be applied for a derivation of constraints on the dynamics of physical systems and of certain types of no-go theorems. For example, time paradoxa can be used against the outcome controllability of entangled subsystems. As a consequence, the requirement of consistency of physical phenomenology induces the “peaceful coexistence” between relativity theory and quantum mechanics. Undecidability in mathematics comes in different varieties; so does undecidability in physics. In physics we have to make sure that the theory is a suitable formal representation of the phenomenology. For example, if the outcome of an experiment cannot be predicted, does that mean that “God plays dice?” Or does it mean that although the causes are in principle known, we are unable to compute a prediction? Or does it simply mean that there are causes but these are unknown to us? These questions may never be fully settled [1], but since Gödel’s [2] and Turing’s [4] centennial findings, remarkable advances have been made in the formal perception of undecidability. And today’s computers not only serve as number crunchers but are becoming a medium to “virtual” realities. This greatly promotes the interaction between theoretical computer sciences, formal logic and the physics of “real” reality. Let us briefly consider the physical correspondents of two forms of mathematical undecidability, the first being associated with the assumption of the