Clarke-Doane, Justin (2025) Logical Dependence of Physical Determinism on Set-theoretic Metatheory. [Preprint]
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Logical dependence of physical determinism on set-theoretic metatheory - 10.17.25 - 2.24am.pdf Download (480kB) |
Abstract
Baroque questions of set-theoretic foundations are widely assumed to be irrelevant to physics. In this article, I demonstrate that this assumption is incorrect. I show that the fundamental physical question of whether a theory is deterministic—whether it fixes a unique future given the present—can depend on one's choice of set-theoretic axiom candidates over which there is intractable disagreement. This dependence is not confined to hypothetical examples. It reaches into mainstream, foundational, and frontier physics, including full discrete systems, the preferred basis problem in quantum mechanics, and the dynamics of Kerr-like black hole interiors. I argue that beyond the familiar analytic notion of well‑posedness, a theory’s determinism profile depends on a regularity layer, on whether the definable sets that carry our ensemble and canonicalization talk are measurable, have the Baire property, and admit measurable selectors. Competing axiom candidates extending ZFC—Gödel’s Axiom of Constructibility (V=L) and large cardinal (LC) assumptions strong enough to imply Projective Determinacy (PD)—diverge on these regularity facts. The divergence has three faces. First, coherence: weak formulations presuppose measurability of coefficients and under V=L one can arrange definable pathologies that collapse the statement of the weak problem, while under PD all projective sets are regular. Second, uniqueness: many determinism results are ensemble claims—“for almost all initial data there is a unique continuation”—whose sense depends on measurability or Baire category at projective complexity. PD secures this, while V=L may not. Third, identity: when multiple admissible continuations remain, physical practice demands a canonical, representation‑independent choice. That demand is a measurable uniformization problem. PD supports measurable, symmetry‑constrained selectors at the Π¹₂ level, while V=L guarantees at most Δ¹₂ (hence possibly non‑measurable) tie‑breaks. I develop a number of live cases, showing how the regularity properties toggle with the metatheory. The upshot is that which extension of standard ZFC we adopt changes what our best‑supported theories say. I close by sketching a research program, reverse physics, on analogy with Friedman’s and Simpson’s reverse mathematics, whose aim is to map a theory’s physical content against the foundational axioms that make its ensemble and canonical claims intelligible. I conclude that, given the entanglement of set-theoretic metatheory and physics, either physical theories must be relativized to set theories (in which case physics itself becomes relative), or, as Quine (1951, 1990) controversially argued, the search for new axioms to settle undecidables may admit of empirical input.
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Logical Dependence of Physical Determinism on Set-theoretic Metatheory. (deposited 15 Aug 2025 15:32)
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