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, we challenge this assumption. We show that even such fundamental questions as whether a theory is deterministic — whether it fixes a unique future given the present — depend on set-theoretic axiom candidates over which there is philosophical 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. In this article, we show, first, that for some mathematically definable physical systems, reasonable set-theoretic assumptions (e.g. ZFC + large cardinals vs. ZFC + V = L) disagree about whether a solution with a particular property exists — such as a solution that's projectively definable. Second, and more dramatically, we show that they disagree about whether any solution to a PDE exists at all. Therefore, contrary to the methodology in physics and its philosophy, determinism is entangled with foundational disputes in set theory.

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Item Type:	Preprint
Creators:	Creators Email ORCID Clarke-Doane, Justin jc4345@columbia.edu
Keywords:	determinism, set theory, v=l, constructibility, philosophy of physics, discrete, projective determinacy, supervenient, mathematical logic, set theory, differential equations,
Subjects:	Specific Sciences > Mathematics > Logic General Issues > Scientific Metaphysics General Issues > Determinism/Indeterminism Specific Sciences > Mathematics Specific Sciences > Physics
Depositing User:	Justin Clarke-Doane
Date Deposited:	16 Jul 2025 15:47
Last Modified:	16 Jul 2025 15:47
Item ID:	25946
Official URL:	https://philpapers.org/go.pl?id=CLALDO-5&proxyId=&...
Subjects:	Specific Sciences > Mathematics > Logic General Issues > Scientific Metaphysics General Issues > Determinism/Indeterminism Specific Sciences > Mathematics Specific Sciences > Physics
Date:	14 July 2025
URI:	https://philsci-archive.pitt.edu/id/eprint/25946

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