Non-Closure Reconstruction: A Structural Reconstruction of Zermelo–Fraenkel Set Theory

This paper develops a reconstruction framework in which the primitive notions are determinations and reconstruction rather than sets and membership. From these primitive concepts, successive layers of structural organization are introduced, including compatibility, stabilization, internal realization, and structural equality. Within this framework, a translation of the language of Zermelo–Fraenkel set theory is defined. The main result shows that the translated Zermelo–Fraenkel axioms are satisfied in the stabilized structural universe obtained from the reconstruction principles. Consequently, classical set theory is interpreted as a stabilized structural layer emerging from a more primitive reconstruction process rather than being taken as the primitive starting point. The paper develops the reconstruction axioms, proves the Structural Reconstruction Theorem, establishes the corresponding interpretability result, and analyzes the logical status of the framework. Representation models are discussed as illustrative mathematical realizations of the reconstruction principles, while questions concerning intrinsic models, relative consistency, and proof-theoretic strength are identified as directions for future research. The work is intended as a contribution to mathematical foundations, proposing an explanatory perspective on the emergence of set-theoretic structure rather than a replacement for classical set theory.