Clarke-Doane, Justin (2025) Logical Dependence of Physical Determinism on Set-theoretic Metatheory. [Preprint]

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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 one’s choice of set-theoretic axiom candidates over which there is intractable disagreement. Suppose, as is customary (Earman 1986), that a deterministic theory is one whose mathematical formulation yields a unique solution to its governing equations. Then the question of whether a physical theory is deterministic becomes the question of whether there exists a unique solution to its mathematical model—typically a system of differential equations. I argue that competing axiom candidates extending standard mathematics—in particular, the Axiom of Constructibility (V = L) and large cardinal axioms strong enough to prove Projective Determinacy—can diverge on all the core dimensions of physical 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 deterministically predicts—can depend on one’s 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 (1951) controversially argued, the search for new axioms to settle undecidables may admit of empirical input.

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Item Type:	Preprint
Creators:	Creators Email ORCID Clarke-Doane, Justin jc4345@columbia.edu
Keywords:	determinism, set theory, pluralism, indeterminism, multiverse, determinacy, incompleteness, descriptive set theory, PDE, Quine, V=L, Axiom of Constructibility, Large Cardinals, new axioms,
Subjects:	Specific Sciences > Mathematics > Applicability Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Explanation Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic General Issues > Determinism/Indeterminism General Issues > Explanation General Issues > Laws of Nature General Issues > Logical Positivism/Logical Empiricism Specific Sciences > Mathematics Specific Sciences > Physics > Quantum Gravity
Depositing User:	Justin Clarke-Doane
Date Deposited:	15 Aug 2025 15:32
Last Modified:	15 Aug 2025 15:32
Item ID:	26235
Subjects:	Specific Sciences > Mathematics > Applicability Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Explanation Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Logic General Issues > Determinism/Indeterminism General Issues > Explanation General Issues > Laws of Nature General Issues > Logical Positivism/Logical Empiricism Specific Sciences > Mathematics Specific Sciences > Physics > Quantum Gravity
Date:	2025
URI:	https://philsci-archive.pitt.edu/id/eprint/26235

Available Versions of this Item

• Logical Dependence of Physical Determinism on Set-theoretic Metatheory. (deposited 15 Aug 2025 15:32) [Currently Displayed]
  • Logical Dependence of Physical Determinism on Set-theoretic Metatheory. (deposited 18 Oct 2025 13:00)

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