📝 SummarXiv

Abstract

Baroque questions of set-theoretic foundations are widely assumed to be irrelevant to physics. In this article, I challenge this assumption. I argue that even the fundamental physical question of whether a theory is deterministic - whether it fixes a unique future given the present - can depend on choice of set-theoretic axiom candidates over which there is deep disagreement. Suppose, as is customary, that a deterministic theory is one whose mathematical formulation yields a unique solution to its governing equations. Then the question of whether a theory is deterministic is the question of whether there exists a unique solution to its mathematical model. I argue that competing axiom candidates extending standard mathematics can diverge on all dimensions of determinism. First, they may disagree about whether a given physical system is well-posed, and so whether a solution exists. Second, even when they agree that a solution exists, they can differ on whether that solution is unique. Finally, even when they agree that a system has a solution, and agree that this solution is unique, they may still dispute what that solution is. Whether a theory is deterministic - and even which outcome it predicts - can depend on choice of set-theoretic metatheory. I indicate how the conclusions extend to discrete systems and suggest directions for future research. One upshot of the discussion is that either physical theories must be relativized to set-theoretic metatheories, in which case physics itself becomes relative, or, as Quine controversially argued, the search for new axioms to settle undecidables may admit of empirical input.