V = L or Large Cardinals? The Metatheoretic Dependence of Physical Determinism, with Application to the Kerr Cauchy Horizon
It is widely assumed that recondite debates about extensions of ZFC — in particular, between V=L and large-cardinal (LC) hypotheses sufficient to imply Projective Determinacy — are irrelevant to physics. I argue that this insularity thesis is false. The dependence comes in two forms that have not been distinguished. At the “analytic layer” (existence, uniqueness, continuous dependence) it can be exhibited for mathematically natural counterfactual systems, in the spirit of Norton’s Dome — coherence, uniqueness, and even the identity of “the” unique solution can toggle between V = L and LC/PD. At the “regularity layer” (robustness, typicality, canonicalization), the dependence may reach into actual physics, including statistical mechanics and the determinism of Kerr black hole interiors. Whether it does turns on a philosophical thesis about how determinism ought to be regimented, a thesis I defend by analogy with general covariance. I present one unconditional must-diverge result (a nearest-neighbor Ising model on ℤ³) and one slow-spin Kerr theorem, and I propose Physical Strong Cosmic Censorship as the reconstructive target whose well-posedness, post-Dafermos–Luk, can itself depend on the V=L/LC dichotomy. I close with the dilemma: either physical theories must be relativized to set-theoretic metatheories, or, as Quine argued, the search for new axioms is continuous with the search for new physical laws.
