The Unrealizable Prescription: Physically Derivable Set Theory as the Replacement for ZFC

Zermelo-Fraenkel set theory with the axiom of choice (ZFC) is the axiomatic foundation of virtually all modern mathematics. This paper argues that ZFC is refuted by the physical construction criterion: a mathematical object exists if and only if it is constructed by a finite physical process. The argument does not rest on a competing mathematical intuition or a preference for the finite. It rests on a necessary consequence of the Minimally Physically Derivable Theories (MPDT) metatheory and its instantiation in Quantum-Geometry Dynamics (QGD): space is constituted by a finite discrete structure of preons(−), and every physical process is bounded by finite resources. Since every mathematical construction is a physical process, and every physical process is finite, no completed infinite mathematical object can exist. Nine of ZFC's ten axioms describe physically executable operations on finite sets and survive intact under this criterion: Extensionality, Empty Set, Pairing, Union, Power Set, Separation, and Replacement are all valid for finite inputs with finitely decidable or computable conditions. Regularity is redundant — it is a theorem of the resulting theory rather than an independent axiom. One axiom — the Axiom of Infinity — asserts the existence of a completed infinite set without construction. Since no finite physical process can produce a completed infinite totality, the Axiom of Infinity is not false but empty: it asserts the existence of an object that cannot in principle be constructed. Because the entire superstructure of ZFC built on the Axiom of Infinity inherits this emptiness, ZFC is refuted as a physically grounded foundation for mathematics. The Axiom of Infinity is the single source of every physically ungrounded result in ZFC: the uncountability hierarchy, the independence of the continuum hypothesis, the Banach-Tarski paradox, and the need for the Axiom of Choice in its full generality all flow from this one empty axiom. A central distinction is drawn between two types of notation purporting to denote infinite mathematical objects. In the first type, the notation is valid shorthand for the convergence behaviour of a finitely approximable process: the limit notation, the summation notation, and the notation for real numbers as limits of rational approximations. These notations are valid because their physical content — the convergence behaviour of finite partial sums or finite approximations — is fully constructible. The completed infinite object the notation appears to denote does not exist independently of the approximation process and does no mathematical work that the approximation process does not already do. In the second type, the notation has no approximation process interpretation: uncountable cardinals, arbitrary choice functions over infinite collections, the power set of the natural numbers. These notations are unrealizable prescriptions for nothing at all — finite inscriptions that instruct a process which cannot be executed by any physical system and which have no finite shadow that does any physical or mathematical work. Every notation purporting to denote a completed infinite mathematical object is therefore an unrealizable prescription. What cannot be recovered is what never existed in the first place. The replacement is Physically Derivable Set Theory (PDST), whose eight axioms are derived from the MPDT physical construction criterion. The foundational axiom is the Physical Construction Axiom: a set exists if and only if it is produced by a finite sequence of physically executable operations applied to previously existing sets, beginning from the empty set. The remaining axioms — Extensionality, Empty Set, Pairing, Union, Power Set, Separation (restricted to finitely decidable properties), and Replacement (restricted to finitely computable functions) — describe the physically executable operations on finite sets. The Axiom of Infinity does not appear. The Axiom of Choice is not an axiom but a theorem for finite collections, derivable from the construction axiom. Regularity is a theorem rather than an axiom. PDST is consistent, complete, and decidable. It is equivalent in strength to the theory of hereditarily finite sets, which is itself equivalent to Peano arithmetic. It encompasses all of finitary mathematics and all of physics. The continuum hypothesis cannot be formulated in PDST. The Banach-Tarski paradox does not arise. Gödelian incompleteness is precluded because the self-referential Gödel construction requires proofs of arbitrary length, which are unavailable in a physically bounded framework. The Axiom of Choice becomes a theorem. Russell's paradox cannot arise because the property "x ∉ x" is not finitely decidable and is therefore excluded by the Separation axiom's finite decidability requirement. What PDST does not recover is what never existed: the completed infinite objects that the Axiom of Infinity introduced by assertion rather than construction. This is not a cost but a clarification. Every theorem in finitary mathematics survives. Every physical prediction survives. Every computable result survives. What is lost is precisely what generated the century of controversy, paradox, and undecidability results that ZFC's single empty axiom produced — and that was never needed for a single physical application.