Finite Describability and the Computational Hierarchy of Physical Systems: Why True Algorithmic Randomness Necessitates a Random Oracle

What does it mean for the universe to be governed by laws? We formalize the notion of a finitely describable physical system within the language of set theory and computability. A hierarchy is established, proving that: (1) any deterministic finitely describable system is exactly simulable by a deterministic Turing machine; (2) any stochastic finitely describable system without a random oracle is simulable by a probabilistic Turing machine to arbitrary precision, or by a deterministic machine using a pseudorandom source; and (3) if such a system produces observation sequences that are algorithmically random (in the sense of Martin-Löf), then its formal description must inherently include a non-computable element—a random oracle. This result provides a rigorous demarcation between mere statistical complexity and genuine algorithmic randomness. Applying this framework to quantum mechanics suggests that, under the standard interpretation, any finite axiomatization must implicitly contain a non-computable random oracle, reframing the question of cosmic determinism as a question of finite describability without oracles.