VR-Sets: An Operational Theory of Sets Grounded in Leibnizian Void

VR-Sets is the third work in the VR cycle, following VR. A Formal System (Zenodo DOI 10.5281/zenodo.20212092) and VR-Numbers (Zenodo DOI 10.5281/zenodo.20272743). It is an operational theory of sets built on a single ontological primitive — the empty operationality ∅ — under the slogan "only ∅ is, all else is doing." A set is identified with its membership functionality: a procedure that, upon query, either reveals an element (itself a set) or reveals nothing. Membership x ∈ y is read as reference, not as containment. The closure principle — every operationally describable functionality is a set — replaces the existence axioms of ZF and dissolves Russell's paradox ontologically: contradictory descriptions simply fail to specify functionalities. Principal results: (1) All nine ZF axioms together with AC are derived as closure theorems or follow directly from the definitions. Extensionality is a tautology of operational identity ≡; the replacement schema collapses into a single theorem; AC becomes a theorem on the countable operational universe. (2) Two ontological modes — ZFC-mode (well-founded functionalities) and ZFA-mode (all describable functionalities) — are defined through well-foundedness as a structural property of a functionality, without temporal ordering of construction. Both modes are non-contradictory. Conjectures of mutual interpretability with countable models of classical ZFC and ZFC− + AFA are stated as open questions. (3) The operational universe is countable. Cantor's diagonal argument fails to construct a new subset because its premise — a complete enumeration of describable subsets — is operationally unrealisable (equivalent to the halting problem). The Banach–Tarski paradox is absent by construction, not by weakening AC. (4) VR numbers from the formal system VR are specific well-founded operational sets, automatically in the ZFC-mode; arithmetic is inherited through direct isomorphism without re-proof. (5) The operationalist position is articulated as a coherent alternative to Platonism, formalism, constructivism, and structuralism — sharing with computable analysis, predicativism, and ZF + DC + AD the restriction to operationally given objects, but arrived at by ontological choice rather than by restriction of a classical universe. Nine parts; the preprint closes by recording open questions — technical (machine formalisation in Lean, proofs of equivalence conjectures), substantive (connections with topoi and homotopy type theory, extensions to MK/NBG), and philosophical (the nature of "action," the next work of the VR cycle).